Day 8 — Exponential Functions, Summation & Gate II

Day 8 — Exponential Functions, Summation & Gate II

Day 20 — Exponential growth & decay

Stage II · concept day
Goal: Model \(a\cdot r^{t}\) growth/decay; compute doubling time and half-life; compare linear vs exponential tables; use compound interest; treat logs as inverses of exponentials to solve for time.

Why this matters

Moore’s-law style growth, epidemic toy models, cache miss penalties that multiply, and compound interest are exponential. Algorithms that are \(2^n\) become infeasible quickly—the table comparison makes this visceral. Logs convert multiply-growth into additive time equations.

Theory

Exponential functions

For base \(b>0\), \(b\neq 1\): \[f(x)=b^{x}.\]

• Domain \(\mathbb{R}\) (for real teaching; constructive details aside).
  
• Range \((0,\infty)\).
  
• If \(b>1\): increasing (growth). If \(0<b<1\): decreasing (decay).
  
• \(b^{0}=1\); \(b^{1}=b\); \(b^{-x}=1/b^{x}\).

General model: \[A(t)=A_0\, r^{t}\] or \(A(t)=A_0 e^{kt}\) with \(k=\ln r\) when using continuous base \(e\).

Often \(r=1+i\) for growth rate \(i\), or \(r=1-\delta\) for decay.

Doubling time

If \(A(t)=A_0\cdot 2^{t/T_d}\), then \(T_d\) is the doubling period.
If \(A(t)=A_0 b^{t}\) with \(b>1\), solve \(b^{T}=2\): \(T=\log_b 2=\dfrac{\ln 2}{\ln b}\).

Half-life

If \(A(t)=A_0\cdot (1/2)^{t/T_h}\), then \(T_h\) is half-life.
For \(A(t)=A_0 e^{-kt}\) (\(k>0\)), \(T_h=\dfrac{\ln 2}{k}\).

Linear vs exponential tables

\(t\)	Linear \(2t+1\)	Exp \(2^{t}\)
0	1	1
1	3	2
2	5	4
3	7	8
4	9	16
5	11	32

Linear: constant additive change. Exponential: constant multiplicative factor.

Rule of thumb: any exponential with base \(>1\) eventually dominates any polynomial (Stage VIII makes precise).

Compound interest

Principal \(P\), annual rate \(r\) (decimal), compounded \(n\) times per year for \(t\) years: \[A=P\biggl(1+\frac{r}{n}\biggr)^{nt}.\]

Continuous compounding: \(A=Pe^{rt}\).

Effective annual rate: \(\bigl(1+\frac{r}{n}\bigr)^{n}-1\).

Logs invert exponentials

\[y=b^{x}\quad\Leftrightarrow\quad x=\log_b y.\]

Solve \(A_0 r^{t}=A\) by \(t=\log_r(A/A_0)=\dfrac{\ln(A/A_0)}{\ln r}\).

Exponential equations (review+)

Same base: \(b^{f(x)}=b^{g(x)}\Rightarrow f(x)=g(x)\).
Different bases: take log, or rewrite with prime bases.

Exponent laws (active recall)

For \(b>0\), \(b\neq 1\) and real exponents as taught: \[ b^{x+y}=b^x b^y,\quad b^{xy}=(b^x)^y,\quad (ab)^x=a^x b^x,\quad b^{-x}=\frac{1}{b^x}. \] Logs invert: \(\log_b(xy)=\log_b x+\log_b y\), \(\log_b(x^r)=r\log_b x\), change of base \(\log_b x=\frac{\ln x}{\ln b}\).

Why exponential beats polynomial (sketch)

Fix \(r>1\) and degree \(d\). Ratios \(\frac{r^{n+1}/(n+1)^d}{r^n/n^d}=r\bigl(\frac{n}{n+1}\bigr)^d\to r>1\), so eventually the sequence \(r^n/n^d\) increases and \(\to\infty\). Thus \(r^n\) grows faster than \(n^d\). (Stage VIII: Big-O formalizes this.)

Compound frequency comparison

For fixed \(P,r,t>0\), the function \(n\mapsto P\bigl(1+\frac{r}{n}\bigr)^{nt}\) increases with \(n\) toward \(Pe^{rt}\). More frequent compounding → slightly more money, with continuous as the upper envelope.

Worked examples

Example 1 — Evaluate

\(3^{4}=81\), \(2^{-3}=\frac{1}{8}\), \((\frac{1}{2})^{3}=\frac{1}{8}\).

Example 2 — Growth model

Bacteria \(A_0=500\), double every \(3\) hours: \(A(t)=500\cdot 2^{t/3}\). At \(t=9\): \(500\cdot 2^{3}=4000\).

Example 3 — Doubling time

\(A=100\cdot (1.05)^{t}\). Doubling: \(1.05^{T}=2\), \(T=\frac{\ln 2}{\ln 1.05}\approx\frac{0.693}{0.0488}\approx 14.2\) years.

Example 4 — Half-life

Isotope half-life \(8\) days, \(A_0=40\) g: \(A(t)=40\cdot(\frac{1}{2})^{t/8}\). After \(24\) days: \(40\cdot\frac{1}{8}=5\) g.

Example 5 — Compound interest

\(P=1000\), \(r=0.06\), \(n=12\), \(t=2\): \(A=1000(1.005)^{24}\approx 1127.49\).

Example 6 — Continuous

\(1000 e^{0.06\cdot 2}=1000 e^{0.12}\approx 1127.5\) (similar).

Example 7 — Solve for time

\(500\cdot 1.03^{t}=800\). \(1.03^{t}=1.6\). \(t=\frac{\ln 1.6}{\ln 1.03}\approx 15.9\).

Example 8 — Linear vs exp

After \(10\) steps: linear \(2\cdot 10+1=21\) vs \(2^{10}=1024\).

Example 9 — Decay factor

Lose \(20\%\) per year: multiply by \(0.8\) each year. After \(5\) years: \(A_0(0.8)^{5}\).

Example 10 — Log inverse

\(f(x)=2^{x}\), \(f^{-1}(x)=\log_2 x\). Compose: \(2^{\log_2 9}=9\).

Example 11 — Effective rate

\(r=12\%\) compounded monthly: \((1.01)^{12}-1\approx 0.1268=12.68\%\) effective.

Example 12 — Equation

\(4^{x}=8^{x-1}\Rightarrow (2^2)^{x}=(2^3)^{x-1}\Rightarrow 2x=3x-3\Rightarrow x=3\).

Example 13 — Table second look

Exponential \(3^{t}\): ratios \(\frac{A(t+1)}{A(t)}=3\) constant.

Example 14 — Bits growth

Address space \(2^{n}\) grows by factor \(2\) when \(n\) increases by \(1\)—exponential in bit width.

Example 15 — Substitution exponential equation

\(2^{2x}-2^{x}-6=0\). Set \(u=2^{x}>0\): \(u^2-u-6=0\), \((u-3)(u+2)=0\), \(u=3\) (reject \(-2\)). \(2^{x}=3\), \(x=\log_2 3\).

Example 16 — Rule of 72

At \(r=0.06\), rule of 72 gives \(72/6=12\) years; exact \(T=\ln 2/\ln 1.06\approx 11.9\). Good approximation.

Example 17 — Half-life to continuous \(k\)

\(T_h=8\) days: \(e^{-k\cdot 8}=\frac12\), \(k=\frac{\ln 2}{8}\). Then \(A(t)=A_0 e^{-(\ln 2)t/8}\).

Example 18 — Comparing early table values

\(100n\) vs \(2^n\): at \(n=1..9\), linear can win; at \(n=10\), \(1000\) vs \(1024\)—already close; \(n=20\): \(2000\) vs \(>10^6\). Never trust small-\(n\) tables alone for asymptotics.

Example 19 — Password entropy

Charset size \(62\), length \(8\): \(62^8\) possibilities; bits \(\log_2(62^8)=8\log_2 62\approx 8\cdot 5.95=47.6\) bits. Each extra character adds \(\log_2 62\) bits.

Example 20 — Same growth two writings

\(A=100\cdot 4^{t}=100\cdot (2^2)^{t}=100\cdot 2^{2t}=100\cdot e^{t\ln 4}\). Discrete base \(4\) per unit time equals continuous \(k=\ln 4\).

Exercises

Easy

1. Evaluate \(5^{3}\), \(10^{-2}\), \((\frac{2}{3})^{2}\).
2. \(A(t)=20\cdot 2^{t}\): find \(A(0)\), \(A(3)\).
3. Write a model: starts at \(100\), triples every period \(t\) in periods.
4. Half-life \(10\) h, start \(80\) mg: amount after \(30\) h.
5. Solve \(2^{x}=32\).

Medium

6. Doubling time for \(6\%\) annual growth (annual compound once): \(1.06^{T}=2\).
7. \(P=2000\), \(r=4\%\), compounded quarterly, \(t=5\) years: write expression for \(A\) (compute if desired).
8. Compare after \(t=8\): \(10+3t\) vs \(10\cdot (1.2)^{t}\) qualitatively which larger.
9. Solve \(300 e^{0.02 t}=900\) for \(t\).
10. If half-life is \(T_h\), show \(A(2T_h)=A_0/4\).
11. Effective annual rate for \(r=0.05\) compounded monthly.
12. Rewrite \(A=50\cdot 4^{t}\) as \(A=50\cdot 2^{?}\).
13. Decay \(15\%\) per step for \(6\) steps: overall factor.

Hard

14. Prove that \(f(x)=b^{x}\) for \(b>1\) is injective; conclude logs well-defined.
15. Derive doubling time formula \(T=\ln 2/\ln b\) for \(A_0 b^{t}\).
16. Show continuous model \(Pe^{rt}\) is limit idea of \((1+\frac{r}{n})^{n}\to e^{r}\) (state without full limit proof).
17. Solve \(2^{2x}-2^{x}-6=0\) with \(u=2^{x}\).
18. Compare polynomial \(n^{5}\) and exponential \(1.1^{n}\) for \(n=10,50,100\) with a calculator after estimating by hand which wins eventually.
19. If \(A(t)=A_0 e^{kt}\) and \(A(3)=2A_0\), find \(k\) and half-life/doubling.
20. Prove \((b^{x})(b^{y})=b^{x+y}\) using exponent laws (Day 8 link).

Challenge / CS-flavored

21. Password entropy: space size \(N\), “bits” \(\log_2 N\). If \(N\) multiplies by \(64\), bits increase by \(?\)
22. Algorithm A: \(1000n\) ops; B: \(2^{n}\). For which \(n\) is B already larger? (try small \(n\) table)
23. Cache: miss penalty multiplies latency by \(100\) each level of memory—exponential cost stack idea.
24. Version growth: semantic version major bumps as rare exponential compatibility breaks—metaphor only.
25. Solve for bits \(n\) in \(2^{n}\ge 10^{12}\).

Continuous growth rate

If \(A=A_0 e^{kt}\), the instantaneous relative growth rate is \(k\) (calculus interpretation later).
For discrete \(A_{n+1}=(1+i)A_n\), relative growth per step is \(i\).

Log scales

Plotting \(y\) on a log scale linearizes pure exponential data: \(\log A(t)=\log A_0 + t\log r\). Slope estimates \(\log r\).

Comparing models

Given data table, look at first differences (linear?) vs ratios \(A_{t+1}/A_t\) (exponential?). Mixed models exist (affine \(A+b\) then exp)—out of scope.

Rule of \(72\) (finance heuristic)

Doubling time years \(\approx 72/(\text{percent rate})\) for modest rates—approximation to \(\ln 2/\ln(1+r)\approx 0.69/r\).

Extra exercises

26. Derive \(T_{1/2}=\ln 2/k\) from \(A_0 e^{-kt}\).
    
27. \(P=5000\) at \(3\%\) continuous for \(10\) years: write \(A\).
    
28. Solve \(4\cdot 3^{x}=36\).
    
29. If compute capacity doubles every \(18\) months, factor over \(6\) years.
    
30. Show \(b^{x}=e^{x\ln b}\) for \(b>0\) (definition bridge).

CS connection

Exponential time algorithms, cryptographic key space \(2^{n}\), compound metrics, and exponential backoff (\(2^{k}\) wait) are daily CS. Logs turn growth into linear equations for “how long until.” Always distinguish \(2^{n}\) from \(n^{2}\).

Common pitfalls

Pitfall	What to do instead
\((1+r)^{t}\) vs \(1+rt\)	Compound vs simple interest
Half-life as subtract half once	Multiplicative \((\frac12)^{t/T}\) each period
Logs of growth without dividing by \(A_0\)	\(t=\log(A/A_0)/\log r\)
Base \(b=1\) as exponential interest	Degenerate constant
Confusing percent and growth factor	\(+5\%\) means \(\times 1.05\)
Claiming linear eventually beats exp	Opposite for base \(>1\)

Fully worked extra examples

E15 — Same-base rewrite.
\((\frac{1}{8})^{x}=2^{x+3}\). Left \((2^{-3})^{x}=2^{-3x}\). So \(-3x=x+3\), \(-4x=3\), \(x=-\frac34\).

E16 — Continuous vs annual.
\(P=1000\), \(r=0.05\), \(t=10\): annual \(1000(1.05)^{10}\); continuous \(1000 e^{0.5}\). Continuous is slightly larger—limit of compounding frequency.

E17 — Half-life chain.
From \(A=A_0(\frac12)^{t/T}\), take log: \(\ln(A/A_0)=-\frac{t}{T}\ln 2\), so \(t=T\frac{\ln(A_0/A)}{\ln 2}\).

E18 — Linear never catches pure exp eventually.
\(1000n\) vs \(2^{n}\): at \(n=20\), \(20000\) vs about \(1\) million. At \(n=10\), \(10000\) vs \(1024\)—linear still larger; crossover later. Always check large \(n\) for exp base \(>1\).

E19 — Compound factor table.
\((1.01)^{12}\approx 1.1268\) (monthly \(12\%\) APR). \((1.12)^{1}=1.12\) simple annual. Difference is compound effect within the year.

End-of-day synthesis problems

S1. Model: start \(200\), grow \(8\%\) per period. Write \(A(t)\); find \(t\) with \(A(t)=500\) in log form.

S2. Half-life \(6\) h, \(A_0=64\): amount at \(t=0,6,12,18,24\).

S3. Compound: \(P=1500\), \(r=0.05\), \(n=4\), \(t=3\) — expression for \(A\).

S4. Table compare \(5+2t\) vs \(5\cdot 2^{t}\) for \(t=0..6\); when does exp overtake?

S5. Solve \(9^{x}=27^{x/2+1}\).

S6. Doubling time for continuous rate \(k=0.07\): \(T=\ln 2/k\).

S7. Effective annual rate for \(12\%\) compounded monthly.

S8. Bits: if key space multiplies by \(2^{10}\), how many bits of security added?

Deep dive — Discrete vs continuous growth

Discrete	Continuous
\(A_{t+1}=r A_t\)	\(A'(t)=k A(t)\) (calculus)
\(A(t)=A_0 r^{t}\)	\(A(t)=A_0 e^{kt}\)
\(r=e^{k}\) bridge	\(k=\ln r\)
Compound \(n\) times/year	\(n\to\infty\) limit \(e^{rt}\)

Solving for time always uses a log: the log is the inverse “how many multiplications.” In CS, exponential backoff, geometric sequences of array capacity (2^k growth), and \(O(2^n)\) algorithms are the same multiplicative structure.

Synthesis

Exponential models multiply by a constant each period. Doubling time and half-life are special level-crossing times solved with logs. Compound interest is discrete exponential; continuous compounding is the \(e^{rt}\) limit. Linear vs exponential tables teach humility about long horizons. Logs undo exponentials—always isolate the exponential first.

S9. Solve \(2^{2x}-5\cdot 2^{x}+4=0\) via \(u=2^{x}\).

S10. Population \(A_0=1000\) grows \(2\%\) per year: years until \(2500\) (log form + approximate).

S11. Show that if capacity doubles every \(18\) months, then in \(6\) years it multiplies by \(2^{4}=16\).

S12. Prove injectivity of \(b^{x}\) for \(b>1\) using: if \(b^{x}=b^{y}\) then \(b^{x-y}=1\) and \(x-y=0\).

Checkpoint

• Write \(A_0 r^{t}\) models for growth/decay
  
• Compute doubling time / half-life setups
  
• Contrast linear vs exponential tables
  
• Apply compound interest formula
  
• Solve for time using logarithms
  
• S1 and S4 qualitative mastery

Write two takeaways in your own words.

Quick reference card

Model	Formula
Discrete growth	\(A_0 r^{t}\)
Doubling period	\(T=\log_r 2\)
Half-life model	\(A_0(1/2)^{t/T_h}\)
Compound interest	\(P(1+r/n)^{nt}\)
Continuous	\(Pe^{rt}\)
Solve for time	\(t=\log_r(A/A_0)\)
Log inverse	\(y=b^{x}\Leftrightarrow x=\log_b y\)

Exp eventually beats linear when base \(>1\); tables can mislead early.

Tomorrow

Day 21 — Summation notation. \(\sum\) notation, index shifts, linearity, closed forms for \(\sum i\), \(\sum i^2\), geometric sums, telescoping, double sums, nested-loop connection.

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Day 21 — Summation notation

Stage II · concept day
Goal: Read and manipulate \(\sum\) notation; shift indices; use linearity; know closed forms for \(\sum i\), \(\sum i^2\), and geometric series; recognize telescoping sums; introduce double sums; connect to nested loops.

Why this matters

Summation is the language of loop costs, discrete averages, series, and algorithm analysis. Closed forms turn \(O(1)+O(2)+\cdots+O(n)\) into \(\Theta(n^2)\). Telescoping appears in potential-method amortized analysis (idea). Double sums are nested loops.

Theory

Notation

\[\sum_{i=a}^{b} f(i)=f(a)+f(a+1)+\cdots+f(b)\] when \(a,b\in\mathbb{Z}\) and \(a\le b\). If \(a>b\), the sum is empty \(=0\) by convention.

• \(i\) is the index (dummy variable).
  
• \(a\) lower limit, \(b\) upper limit.
  
• \(f(i)\) summand.

\[\sum_{i=1}^{n} i = 1+2+\cdots+n.\]

Index shifts

\[\sum_{i=1}^{n} f(i)=\sum_{k=0}^{n-1} f(k+1)=\sum_{j=2}^{n+1} f(j-1).\]

Reindex carefully: when \(i\to i+1\), limits shift by the same amount.

Linearity

For constants \(\alpha,\beta\): \[\sum_{i=a}^{b}\bigl(\alpha f(i)+\beta g(i)\bigr)=\alpha\sum_{i=a}^{b} f(i)+\beta\sum_{i=a}^{b} g(i).\]

Proof. Distribute addition in the finite expanded sum. \(\square\)

Also: \(\sum_{i=a}^{b} c = c\cdot (b-a+1)\) for constant \(c\) (number of terms).

Closed form: \(\sum i\)

Theorem. \[\sum_{i=1}^{n} i=\frac{n(n+1)}{2}.\]

Proof (Gauss pairing). \((1+n)+(2+(n-1))+\cdots\) gives \(n/2\) pairs each summing to \(n+1\) if \(n\) even; similar if odd.

Proof (induction). Base \(n=1\): \(1=1\). Inductive: \(\frac{n(n+1)}{2}+(n+1)=\frac{(n+1)(n+2)}{2}\). \(\square\)

Closed form: \(\sum i^2\)

Theorem. \[\sum_{i=1}^{n} i^2=\frac{n(n+1)(2n+1)}{6}.\]

Proof. Induction, or from \((i+1)^3-i^3=3i^2+3i+1\) summed (telescoping cubic). \(\square\)

Geometric series

For \(r\neq 1\): \[\sum_{i=0}^{n} r^{i}=\frac{r^{n+1}-1}{r-1}=\frac{1-r^{n+1}}{1-r}.\]

Proof. Let \(S=\sum_{i=0}^{n} r^{i}\). Then \(rS=\sum_{i=1}^{n+1} r^{i}\), so \(S-rS=1-r^{n+1}\). \(\square\)

If \(|r|<1\), infinite sum \(\sum_{i=0}^{\infty} r^{i}=\frac{1}{1-r}\) (Stage later); finite form always primary today.

Special: \(\sum_{i=0}^{n} 2^{i}=2^{n+1}-1\).

Telescoping sums

If summand is a difference \(f(i)-f(i+1)\) or \(f(i+1)-f(i)\): \[\sum_{i=1}^{n}\bigl(f(i)-f(i+1)\bigr)=f(1)-f(n+1).\]

Classic: \(\dfrac{1}{i(i+1)}=\dfrac{1}{i}-\dfrac{1}{i+1}\) (partial fractions Day 17).

\[\sum_{i=1}^{n}\frac{1}{i(i+1)}=1-\frac{1}{n+1}=\frac{n}{n+1}.\]

Double sums

\[\sum_{i=1}^{m}\sum_{j=1}^{n} a_{ij}=\sum_{j=1}^{n}\sum_{i=1}^{m} a_{ij}\] for finite sums (absolute convergence automatic for finite).

If \(a_{ij}=f(i)g(j)\) separates: \(\bigl(\sum_i f(i)\bigr)\bigl(\sum_j g(j)\bigr)\).

Region sums: \(\sum_{i=1}^{n}\sum_{j=1}^{i} 1=\sum_{i=1}^{n} i=\frac{n(n+1)}{2}\) — triangle of pairs \((i,j)\) with \(1\le j\le i\le n\).

Connection to nested loops

count = 0
for i = 1..n:
  for j = 1..i:
    count += 1

runs \(\sum_{i=1}^{n} i = n(n+1)/2\) times.

Outer \(n\), inner \(m\) independent: \(nm\) iterations \(=\sum_{i=1}^{n}\sum_{j=1}^{m} 1\).

Worked examples

Example 1 — Expand

\(\sum_{k=3}^{6} (2k-1)=5+7+9+11=32\).

Example 2 — Linearity

\(\sum_{i=1}^{n}(3i+2)=3\cdot\frac{n(n+1)}{2}+2n=\frac{3n(n+1)+4n}{2}\).

Example 3 — Shift

\(\sum_{i=2}^{n} i=\bigl(\sum_{i=1}^{n} i\bigr)-1=\frac{n(n+1)}{2}-1\).

Example 4 — Geometric

\(\sum_{i=0}^{5} 3^{i}=\frac{3^{6}-1}{3-1}=\frac{729-1}{2}=364\).

Example 5 — Geometric factor

\(\sum_{i=1}^{n} 2^{i}=2\frac{2^{n}-1}{2-1}=2^{n+1}-2\).

Example 6 — Telescoping

\(\sum_{i=1}^{n}\bigl(\frac{1}{i}-\frac{1}{i+1}\bigr)=1-\frac{1}{n+1}\).

Example 7 — \(i^2\)

\(\sum_{i=1}^{10} i^2=\frac{10\cdot 11\cdot 21}{6}=385\).

Example 8 — Double

\(\sum_{i=1}^{2}\sum_{j=1}^{3} (i+j)=\sum_{i=1}^{2}\bigl((i+1)+(i+2)+(i+3)\bigr)=\sum_{i=1}^{2}(3i+6)=3\sum i+12=9+12=21\) for \(i\) to \(2\): \(3(1+2)+2\cdot 6=9+12=21\).

Example 9 — Triangle

\(\sum_{i=1}^{n}\sum_{j=1}^{i} j=\sum_{i=1}^{n}\frac{i(i+1)}{2}=\frac{1}{2}\sum(i^2+i)=\frac{1}{2}\bigl(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}\bigr)\).

Example 10 — Factor out

\(\sum_{i=1}^{n} 5=5n\).

Example 11 — Change index

\(\sum_{k=0}^{n-1} (k+1)=\sum_{i=1}^{n} i=\frac{n(n+1)}{2}\).

Example 12 — Loop cost

for i in 1..n: for j in 1..n: O(1) → \(n^2\) ops.
for i in 1..n: for j in 1..i: O(1) → \(n(n+1)/2\).

Example 13 — Arithmetic series

\(\sum_{i=0}^{n-1} (a+id)=n a + d\frac{(n-1)n}{2}\).

Example 14 — Finite geometric decay

\(\sum_{i=0}^{n} \bigl(\frac{1}{2}\bigr)^{i}=\frac{1-(1/2)^{n+1}}{1-1/2}=2\bigl(1-2^{-(n+1)}\bigr)<2\).

Exercises

Easy

1. Expand \(\sum_{i=1}^{4} i^{2}\).
2. Compute \(\sum_{i=1}^{100} 1\).
3. Compute \(\sum_{i=1}^{n} 4\) in closed form.
4. Evaluate \(\sum_{i=0}^{3} 2^{i}\).
5. Write \(1+3+5+7\) in \(\sum\) notation.

Medium

6. Prove \(\sum_{i=1}^{n} i=\frac{n(n+1)}{2}\) by induction.
7. Compute \(\sum_{i=1}^{n} (2i-1)\) (odd numbers); simplify closed form.
8. Compute \(\sum_{i=0}^{n} 5\cdot 3^{i}\).
9. Reindex \(\sum_{i=3}^{n} (i-2)\) to start at \(j=1\).
10. Telescoping: \(\sum_{i=1}^{n}\bigl(\frac{1}{2i-1}-\frac{1}{2i+1}\bigr)\).
11. Double sum \(\sum_{i=1}^{n}\sum_{j=1}^{m} 1\).
12. \(\sum_{i=1}^{n}\sum_{j=1}^{i} 1\).
13. Use \(\sum i^{2}\) to get \(\sum_{i=1}^{n} (i+1)^{2}\) with a shift.

Hard / proof

14. Prove the geometric sum formula for \(r\neq 1\).
15. Prove \(\sum_{i=1}^{n} i^{2}=\frac{n(n+1)(2n+1)}{6}\) by induction.
16. From \((i+1)^{3}-i^{3}=3i^{2}+3i+1\), sum and derive the \(i^{2}\) formula (assuming \(\sum i\) known).
17. Show \(\sum_{i=1}^{n}\frac{1}{i(i+1)}=\frac{n}{n+1}\).
18. Prove linearity for finite sums from definition.
19. Evaluate \(\sum_{1\le j\le i\le n} 1\) two ways.
20. Find closed form \(\sum_{i=1}^{n} i\cdot r^{i}\) for \(r\neq 1\) (differentiate geometric or formula \(r\frac{1-(n+1)r^{n}+nr^{n+1}}{(1-r)^{2}}\)).

Challenge / CS-flavored

21. Nested loops: \(i=0..n-1\), \(j=i..n-1\) — express iteration count as a sum and closed form.
22. Cost \(\sum_{i=1}^{n} \Theta(i)=\Theta(n^{2})\) — justify using sum bounds (optional sandwich).
23. Binary heap height ideas: \(\sum_{k=0}^{h} 2^{k}=2^{h+1}-1\) nodes in perfect tree.
24. Prefix sums: \(S_k=\sum_{i=1}^{k} a_i\) — write \(a_k\) as telescoping \(S_k-S_{k-1}\).
25. Compare \(\sum_{i=1}^{n} \log i\) vs \(n\log n\) qualitatively (integral intuition OK).

Summation bounds (algorithm analysis seed)

For increasing \(f\), \[\int_{0}^{n} f(x)\,dx \le \sum_{i=1}^{n} f(i) \le f(n)+\int_{0}^{n-1} f(x)\,dx\] (integral test pictures)—optional calculus. Cruder: \(n\cdot \min f \le \sum f \le n\cdot\max f\) on finite ranges.

Arithmetic series general

First term \(a\), common difference \(d\), \(n\) terms: \[S=\frac{n}{2}\bigl(2a+(n-1)d\bigr)=\frac{n}{2}(\text{first}+\text{last}).\]

Empty and singleton sums

\(\sum_{i=1}^{0}=0\) empty. \(\sum_{i=5}^{5} f(i)=f(5)\).

Changing order in dependent double sums

\(\sum_{i=1}^{n}\sum_{j=1}^{i} a_{ij}=\sum_{j=1}^{n}\sum_{i=j}^{n} a_{ij}\) by switching the triangle region \(1\le j\le i\le n\).

Extra exercises

26. Prove arithmetic series formula by pairing or induction.
    
27. Evaluate \(\sum_{i=1}^{n} (i^{2}-i)\).
    
28. Switch order: \(\sum_{i=1}^{n}\sum_{j=i}^{n} 1\).
    
29. Closed form \(\sum_{i=0}^{n} (2i+1)= (n+1)^{2}\) (odds).
    
30. Cost of for i=1..n: for j=1..i: for k=1..j: O(1) as a triple sum; simplify to \(\binom{n+2}{3}\) or \(\frac{n(n+1)(n+2)}{6}\).

CS connection

Runtime of nested loops is double sums. Series closed forms give exact operation counts. Geometric sums model binary trees, geometric backoff totals, and residual geometric series in hashing analyses. Telescoping is the discrete fundamental theorem of calculus.

Common pitfalls

Pitfall	What to do instead
Off-by-one limits	Count terms: \(b-a+1\)
Forgetting \(i=0\) term in geometric	Check lower limit
\(\sum i = n^{2}/2\) without \(+n/2\)	Use \(\frac{n(n+1)}{2}\)
Moving index without shifting limits	Always adjust bounds
Infinite geometric with \(|r|\ge 1\)	Finite formula only unless \(|r|<1\)
Nested loop both to \(n\) as \(n\) not \(n^{2}\)	Independent loops multiply

Fully worked extra examples

E15 — Linearity combo.
\(\sum_{i=1}^{n}(i^{2}+3i+2)=\frac{n(n+1)(2n+1)}{6}+3\cdot\frac{n(n+1)}{2}+2n\). Factor \(n\) if desired.

E16 — Geometric application.
Binary tree nodes levels \(0..h\): \(\sum_{k=0}^{h}2^{k}=2^{h+1}-1\).

E17 — Telescoping partial fractions.
\(\sum_{i=1}^{n}\frac{1}{i(i+1)}=\sum_{i=1}^{n}\bigl(\frac{1}{i}-\frac{1}{i+1}\bigr)=1-\frac{1}{n+1}\).

E18 — Double sum separate.
\(\sum_{i=1}^{m}\sum_{j=1}^{n} ij=\bigl(\sum_{i=1}^{m} i\bigr)\bigl(\sum_{j=1}^{n} j\bigr)=\frac{m(m+1)}{2}\cdot\frac{n(n+1)}{2}\).

E19 — Triangle count two ways.
Number of pairs \(1\le j\le i\le n\) equals \(\sum_{i=1}^{n} i=\binom{n+1}{2}\).

End-of-day synthesis problems

S1. Prove \(\sum_{i=1}^{n} i=\frac{n(n+1)}{2}\) by induction with base and inductive step written fully.

S2. Closed form \(\sum_{i=1}^{n} (4i-1)\).

S3. Geometric \(\sum_{i=0}^{n} (\frac{1}{3})^{i}\).

S4. Telescoping \(\sum_{i=1}^{n} \bigl(\frac{1}{i}-\frac{1}{i+2}\bigr)\) (partial expand first terms/last).

S5. Double: \(\sum_{i=1}^{n}\sum_{j=1}^{i} j\) simplified.

S6. Loop for i=1..n: for j=i..n: cost 1 — exact count.

S7. Reindex \(\sum_{i=0}^{n-1} (i+1)^{2}\) into a sum starting at \(1\).

S8. Prove geometric sum formula via \(S-rS\).

Checkpoint

• Expand and reindex sums
  
• Apply linearity and constant sums
  
• Use \(\sum i\), \(\sum i^{2}\), geometric closed forms
  
• Recognize and evaluate telescoping sums
  
• Translate nested loops into double sums
  
• S1 and S8 as must-pass proofs

Write two takeaways in your own words.

Deep dive — sums, closed forms, double sums, loops

D1 — Induction for \(\sum i\) fully written.
Claim \(P(n):\sum_{i=1}^{n}i=\frac{n(n+1)}{2}\).
Base \(n=1\): \(1=1\).
Inductive: assume \(P(n)\); then \(\sum_{i=1}^{n+1}i=\frac{n(n+1)}{2}+(n+1)=\frac{n(n+1)+2(n+1)}{2}=\frac{(n+1)(n+2)}{2}\).

D2 — Linearity combo closed form.
\(\sum_{i=1}^{n}(4i-1)=4\cdot\frac{n(n+1)}{2}-n=2n(n+1)-n=n(2n+2-1)=n(2n+1)\).

D3 — Geometric sum via \(S-rS\).
\(S=\sum_{i=0}^{n}r^{i}\), \(rS=\sum_{i=1}^{n+1}r^{i}\), \(S-rS=1-r^{n+1}\), so \(S=\frac{1-r^{n+1}}{1-r}\) for \(r\neq 1\).
Special: \(\sum_{i=0}^{n}2^{i}=2^{n+1}-1\).

D4 — Telescoping with gap \(2\).
\(\sum_{i=1}^{n}\bigl(\frac{1}{i}-\frac{1}{i+2}\bigr)=\bigl(1+\frac12+\frac13+\cdots+\frac{1}{n}\bigr)-\bigl(\frac{1}{3}+\cdots+\frac{1}{n+2}\bigr)=1+\frac12-\frac{1}{n+1}-\frac{1}{n+2}\).

D5 — Triangle double sum.
\(\sum_{i=1}^{n}\sum_{j=1}^{i}1=\sum_{i=1}^{n}i=\frac{n(n+1)}{2}\).
\(\sum_{i=1}^{n}\sum_{j=1}^{i}j=\sum_{i=1}^{n}\frac{i(i+1)}{2}=\frac12\sum(i^{2}+i)=\frac12\Bigl(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}\Bigr)\).

D6 — Nested loop exact counts.
for i=1..n: for j=1..n → \(n^{2}\).
for i=1..n: for j=1..i → \(n(n+1)/2\).
for i=1..n: for j=i..n → also \(n(n+1)/2\) (same triangle, reindexed).

D7 — Separable double sum.
\(\sum_{i=1}^{m}\sum_{j=1}^{n}ij=\bigl(\sum_{i=1}^{m}i\bigr)\bigl(\sum_{j=1}^{n}j\bigr)=\frac{m(m+1)n(n+1)}{4}\).

Extra practice set

31. Closed form \(\sum_{i=1}^{n}(i^{2}+3i+2)\).
    
32. Evaluate \(\sum_{i=0}^{n}(\frac13)^{i}\).
    
33. Switch order: \(\sum_{i=1}^{n}\sum_{j=i}^{n}1\); simplify.
    
34. Prove geometric formula for \(r\neq 1\) from scratch.
    
35. Triple loop i=1..n, j=1..i, k=1..j: write as a sum; recognize \(\frac{n(n+1)(n+2)}{6}\).

Tomorrow

Day 22 — Gate II. Mixed algebra/functions/\(\sum\) portfolio; include a \(2\times 2\) matrix system and a \(\sum\) closed form; sections with answer keys for half.

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Day 22 — Gate II

Stage II · gate day
Goal: Closed-book fluency across Days 11–21: expressions, linear equations/inequalities, \(2\times 2\) systems (including matrix form), polynomials, factoring, rational expressions, functions, linear/quadratic graphs, exponentials, and \(\sum\) notation with closed forms.

Why this matters

Stage III begins logic and proofs. That stage assumes you can manipulate algebra without it consuming all working memory. Gate II certifies that variables, equations, functions, and sums are tools — not obstacles.

Theory

Coverage map

Days	Skills
11	Expressions, distribute, substitute, identities
12	Linear equations; identity vs contradiction
13	Inequalities; intervals; flip rule; absolute value
14	\(2\times 2\) systems; \(\det\); \(Ax=b\); Cramer; inverse
15–16	Polynomials; Horner; factoring; zero-product
17	Rational expressions; domain; partial fractions setup
18	Functions; domain/range; composition; inverse idea
19	Slope, vertex, quadratic formula, complete square
20	Exponential growth/decay; solve \(b^{x}=c\)
21	\(\sum\), linearity, \(\sum i\), geometric, telescoping, double sums

How to sit the gate

1. Write warm-up definitions closed book (below).
   
2. Sections A–D timed if you like (\(100\)–\(140\) min total).
   
3. Section proofs require justifications.
   
4. Log misses by day number; repair within a week.

Warm-up definitions (blank paper)

• Zero-product property
  
• Domain of \(\dfrac{p(x)}{q(x)}\)
  
• Slope formula; vertex \(x=-\dfrac{b}{2a}\)
  
• Function \(f:A\to B\); injective / inverse idea
  
• \(\sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}\)
  
• Geometric sum \(\sum_{i=0}^{n} r^{i}=\dfrac{1-r^{n+1}}{1-r}\) (\(r\neq 1\))
  
• \(\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\)
  
• Solution cases for a \(2\times 2\) linear system
  
• Flip rule for inequalities

Algebra hygiene checklist

• Domain before solving rationals
  
• Flip inequality when multiplying by negative
  
• Check system solutions in both equations
  
• Factor completely before canceling
  
• Expand to verify factoring
  
• Off-by-one on sum limits
  
• \(\det=0\) needs consistency check, not automatic “no solution”
  
• Inverse exists iff \(\det\neq 0\)

Section map

Section	Focus	Suggested time
A	Expressions, linear eq/ineq, absolute value	\(25\) min
B	Systems \(2\times 2\), matrices, polynomials, factoring	\(30\) min
C	Rational, functions, linear/quadratic, exponential	\(30\) min
D	Summation + mixed proofs	\(30\)–\(40\) min

Worked examples (calibration)

Example 1 — Equation + inequality

Solve \(3(x-2)=2x+1\) → \(3x-6=2x+1\) → \(x=7\).
Solve \(-2x+4\le 10\) → \(-2x\le 6\) → \(x\ge -3\), interval \([-3,\infty)\).

Example 2 — System + matrix

\(\begin{cases}x+y=5\\ 2x-y=4\end{cases}\) → add: \(3x=9\), \(x=3\), \(y=2\).
\(A=\begin{pmatrix}1&1\\2&-1\end{pmatrix}\), \(\det=-1-2=-3\neq 0\).
Cramer: \(x=\frac{\det\begin{pmatrix}5&1\\4&-1\end{pmatrix}}{-3}=\frac{-5-4}{-3}=3\).

Example 3 — Inverse

\(A^{-1}=-\frac{1}{3}\begin{pmatrix}-1&-1\\ -2&1\end{pmatrix}=\frac{1}{3}\begin{pmatrix}1&1\\ 2&-1\end{pmatrix}\).
Check \(A A^{-1}=I\).

Example 4 — Factor & rational

\(\dfrac{x^{2}-9}{x^{2}-x-6}=\dfrac{(x-3)(x+3)}{(x-3)(x+2)}=\dfrac{x+3}{x+2}\), \(x\neq 3,-2\).

Example 5 — Quadratic

\(f(x)=x^{2}-4x-5=(x-5)(x+1)\); vertex at \(x=2\), \(f(2)=-9\).

Example 6 — Exponential + sum

\(2^{x}=32\Rightarrow x=5\).
\(\sum_{i=1}^{n}(2i)=2\cdot\frac{n(n+1)}{2}=n(n+1)\).

Example 7 — Function

\(f(x)=\frac{1}{x-1}\), domain \(x\neq 1\); \((f\circ f)(x)=\frac{x-1}{1-(x-1) wait}\): \(f(f(x))=1/(\frac{1}{x-1}-1)=\frac{x-1}{1-(x-1)}=\frac{x-1}{2-x}\).

Example 8 — Geometric sum

\(\sum_{i=0}^{n} 2^{i}=2^{n+1}-1\).

Exercises — Section A (Expressions, linear, inequalities)

Starred items have answer sketches below.

A1.* Simplify \(3(2x-1)-2(x+4)+x\).

A2. Expand \((x-3)(x+5)-x(x-1)\).

A3.* Solve \(4(x-1)=2x+10\); check.

A4. Classify \(3(x+2)=3x+6\) and \(3(x+2)=3x+5\).

A5. Solve \(\dfrac{x}{2}-\dfrac{x}{5}=3\).

A6.* Solve \(-3\le 2x+1<7\); interval notation.

A7. Solve \(|2x-3|\ge 5\); write as union of intervals.

A8. Solve \(|x+1|<0.5\).

A9. Word: three consecutive integers sum to \(75\). Find them.

A10. Rearrange \(A=\frac{1}{2}h(b_1+b_2)\) for \(b_1\).

Exercises — Section B (Systems, poly, factor)

B1.* Solve \(\begin{cases}2x+y=7\\ x-2y=-4\end{cases}\) by elimination; write \(Ax=b\) and \(\det A\).

B2. Use Cramer’s rule on B1 (verify).

B3.* Find \(A^{-1}\) for \(A=\begin{pmatrix}2&1\\ 1&1\end{pmatrix}\); solve \(Ax=\begin{pmatrix}1\\ 0\end{pmatrix}\).

B4. Classify: \(\begin{cases}x+2y=3\\ 2x+4y=6\end{cases}\) and \(\begin{cases}x+2y=3\\ 2x+4y=7\end{cases}\).

B5. Multiply \((x^{2}+1)(x-2)\); degree of product.

B6.* Horner-evaluate \(2x^{3}-3x+1\) at \(x=2\).

B7. Factor \(6x^{2}-7x-3\); solve \(6x^{2}-7x-3=0\).

B8. Factor \(x^{3}-8\); solve \(x^{3}=8\).

B9. Factor completely over reals: \(x^{4}-1\).

B10. Divide \(x^{3}-2x^{2}-x+2\) by \(x-1\) (long division); write factorization if remainder \(0\).

Exercises — Section C (Rational, functions, quad, exp)

C1.* Domain and simplify \(\dfrac{x^{2}-x-6}{x^{2}-9}\).

C2. Compute \(\dfrac{2}{x-1}-\dfrac{1}{x}\); single fraction.

C3.* Partial fractions: \(\dfrac{3x+5}{(x+1)(x+2)}=\dfrac{A}{x+1}+\dfrac{B}{x+2}\); find \(A,B\).

C4. Solve \(\dfrac{x}{x-2}=3\); check domain.

C5. \(f(x)=x^{2}+1\), \(g(x)=x-1\): compute \((f\circ g)(x)\) and \((g\circ f)(x)\).

C6.* Inverse of \(f(x)=5x-2\) on \(\mathbb{R}\).

C7. Vertex form of \(f(x)=x^{2}-6x+4\) by completing the square; vertex coordinates.

C8. Discriminant and roots of \(2x^{2}-3x-2=0\).

C9. Line through \((1,2)\) perpendicular to \(y=2x+3\).

C10.* \(A(t)=100\cdot (1.05)^{t}\); solve \(A(t)=200\) for \(t\) in log form. Half-life style: if decay \(A=80\cdot(1/2)^{t/10}\), find \(A(30)\).

Exercises — Section D (Summation + proofs)

D1.* Closed form: \(\sum_{i=1}^{n}(3i+1)\).

D2. Evaluate \(\sum_{i=0}^{6} 2^{i}\).

D3.* Telescoping: \(\sum_{i=1}^{n}\bigl(\frac{1}{i}-\frac{1}{i+1}\bigr)\).

D4. Double sum: \(\sum_{i=1}^{n}\sum_{j=1}^{i} 1\).

D5. Nested loop iterations: outer \(i=1..n\), inner \(j=1..n\) vs inner \(j=1..i\).

D6.* Prove \(\sum_{i=1}^{n} i=\frac{n(n+1)}{2}\) by induction.

D7. Prove: product of two odds is odd.

D8. Prove flip: if \(c<0\) and \(a<b\) then \(ac>bc\).

D9. Prove \(\det\neq 0\) \(\Rightarrow\) unique solution idea for \(2\times 2\) (via inverse or Cramer).

D10. Geometric: prove \(\sum_{i=0}^{n} r^{i}=\frac{1-r^{n+1}}{1-r}\) for \(r\neq 1\).

Answer key sketches (starred + selected half)

A1. \(6x-3-2x-8+x=5x-11\).
A3. \(4x-4=2x+10\), \(2x=14\), \(x=7\).
A6. \(-4\le 2x<6\), \(-2\le x<3\), \([-2,3)\).

B1. From second \(x=2y-4\); \(2(2y-4)+y=7\), \(5y-8=7\), \(y=3\), \(x=2\). \(A=\begin{pmatrix}2&1\\1&-2\end{pmatrix}\), \(\det=-4-1=-5\).
B3. \(\det=1\), \(A^{-1}=\begin{pmatrix}1&-1\\ -1&2\end{pmatrix}\). \(x=A^{-1}\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}1\\ -1\end{pmatrix}\).
B6. Horner coeffs \(2,0,-3,1\): \(2\); \(4\); \(8-3=5\); \(10+1=11\).

C1. \(\dfrac{(x-3)(x+2)}{(x-3)(x+3)}=\dfrac{x+2}{x+3}\), \(x\neq\pm 3\). (Also \(x\neq 3\) from cancel.)
C3. \(3x+5=A(x+2)+B(x+1)\). \(x=-1\): \(2=A(1)\), \(A=2\). \(x=-2\): \(-1=B(-1)\), \(B=1\).
C6. \(f^{-1}(x)=\frac{x+2}{5}\).
C10. \(t=\log_{1.05}2=\frac{\ln 2}{\ln 1.05}\); \(A(30)=80\cdot\frac{1}{8}=10\).

D1. \(3\cdot\frac{n(n+1)}{2}+n=\frac{3n(n+1)+2n}{2}=\frac{n(3n+5)}{2}\).
D3. \(1-\frac{1}{n+1}=\frac{n}{n+1}\).
D6. Standard induction as Day 21.

Additional half-key:
A4. Identity; contradiction.
A7. \(2x-3\le -5\) or \(2x-3\ge 5\) → \(x\le -1\) or \(x\ge 4\).
B4. Infinite (dependent); empty (inconsistent).
B7. \((3x+1)(2x-3)\); \(x=-\frac13\) or \(x=\frac32\).
C7. \((x-3)^{2}-5\); vertex \((3,-5)\).
C8. \(\Delta=9+16=25\); \(x=2\), \(x=-\frac12\).
D2. \(2^{7}-1=127\).
D4. \(\frac{n(n+1)}{2}\).
D5. \(n^{2}\) vs \(n(n+1)/2\).

Optional rapid-fire

RF1. Combine \(2(x-3)-4(1-x)\).
RF2. Solve \(|x-2|\le 3\); interval.
RF3. \(\det\begin{pmatrix}4&1\\ 2&3\end{pmatrix}\) and inverse factor \(1/\det\).
RF4. Factor \(x^{2}-9x+20\); roots.
RF5. Domain of \(\frac{x}{x^{2}-1}\).
RF6. Vertex of \(y=x^{2}+2x-3\).
RF7. \(3^{x}=81\).
RF8. \(\sum_{i=1}^{n} 2=\;?\)
RF9. \(\sum_{i=0}^{4} 3^{i}\).
RF10. Is \(f(x)=x^{3}\) invertible on \(\mathbb{R}\)?

RF answers: \(6x-10\); \([-1,5]\); \(\det=10\), \(A^{-1}=\frac{1}{10}\begin{pmatrix}3&-1\\ -2&4\end{pmatrix}\); \((x-4)(x-5)\); \(x\neq\pm 1\); \((-1,-4)\); \(x=4\); \(2n\); \(\frac{3^{5}-1}{2}=121\); yes, \(f^{-1}=\sqrt[3]{\cdot}\).

Must-pass items (certify Stage II)

You should be able to, without notes:

1. Solve a \(2\times 2\) system two ways and via \(A^{-1}\) or Cramer once.
   
2. Produce \(\sum_{i=1}^{n} i\) and a geometric sum closed form with proof sketch.
   
3. Complete the square for one quadratic.
   
4. Simplify one rational expression with domain.
   
5. Flip an inequality correctly under negative multiplication.

Repair schedule template

Weak day	Symptom	Redo plan	Date

Self-score rubric

Score band	Meaning
\(\ge 85\%\)	Stage III ready; spot-repair only
\(70\)–\(84\%\)	Repair weak days with targeted rework
\(50\)–\(69\%\)	Re-work failed sections with notes, then retest
\(<50\%\)	Restart Stage II foundations (Days 11–14 first)

CS connection

Gate II underwrites asymptotic sums, solving for parameters in models, matrix transforms in 2D, and function composition pipelines. Stage III will quantify statements about these objects.

Common pitfalls (exam mode)

Pitfall	What to do instead
Flip forget	Mark every \(\times\) by negative
Cancel terms not factors	Factor first
Extraneous rational roots	Check domain
\(\det=0\) auto empty	Test consistency
Sum off-by-one	Count terms \(b-a+1\)
Inverse vs reciprocal	\(f^{-1}\) undoes \(f\)
Losing root dividing by \(x\)	Factor + zero-product

Checkpoint

• Warm-up definitions written closed-book
  
• Sections A–D attempted
  
• At least one \(2\times 2\) matrix system solved (B1–B3)
  
• At least one \(\sum\) closed form derived (D1–D4)
  
• Starred answers checked; miss log by day number
  
• Repair plan dated

Write two takeaways and your three weakest Stage II topics.

Deep dive — mixed Gate II drills with sketches

G1 — Expression → equation → inequality.
Simplify \(3(2x-1)-2(x+4)+x=5x-11\).
Solve \(4(x-1)=2x+10\Rightarrow x=7\).
Solve \(-3\le 2x+1<7\Rightarrow -2\le x<3\), interval \([-2,3)\).
Absolute value: \(|2x-3|\ge 5\Rightarrow x\le -1\) or \(x\ge 4\).

G2 — System + matrix trifecta.
\(\begin{cases}2x+y=7\\ x-2y=-4\end{cases}\): \(x=2\), \(y=3\) (elimination).
\(A=\begin{pmatrix}2&1\\1&-2\end{pmatrix}\), \(\det=-5\).
\(A=\begin{pmatrix}2&1\\1&1\end{pmatrix}\) has \(\det=1\), \(A^{-1}=\begin{pmatrix}1&-1\\ -1&2\end{pmatrix}\); \(A^{-1}\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}1\\ -1\end{pmatrix}\).

G3 — Factor, rational, domain.
\(\dfrac{x^{2}-x-6}{x^{2}-9}=\dfrac{(x-3)(x+2)}{(x-3)(x+3)}=\dfrac{x+2}{x+3}\) for \(x\neq\pm 3\).
Partial fractions setup: \(\dfrac{3x+5}{(x+1)(x+2)}=\dfrac{2}{x+1}+\dfrac{1}{x+2}\).

G4 — Quadratic + exponential.
\(x^{2}-6x+4=(x-3)^{2}-5\), vertex \((3,-5)\).
\(2x^{2}-3x-2=0\): \(\Delta=25\), roots \(2\) and \(-\frac12\).
\(A(t)=100\cdot(1.05)^{t}=200\Rightarrow t=\log_{1.05}2\).

G5 — Summation must-pass.
\(\sum_{i=1}^{n}(3i+1)=\frac{n(3n+5)}{2}\).
\(\sum_{i=0}^{6}2^{i}=127\).
Telescoping \(\sum_{i=1}^{n}\bigl(\frac1i-\frac1{i+1}\bigr)=\frac{n}{n+1}\).
Double \(\sum_{i=1}^{n}\sum_{j=1}^{i}1=\frac{n(n+1)}{2}\).

G6 — Proof fragments to rewrite cold.
- Induction for \(\sum i\).
- Geometric \(S-rS\).
- Flip rule: \(c<0\), \(a<b\Rightarrow ac>bc\).
- \(\det\neq 0\Rightarrow\) unique \(2\times 2\) solution via inverse/Cramer.

Extra rapid-fire (second pass)

RF11. Solve \(|x-2|\le 3\); interval.
RF12. Factor \(x^{2}-9x+20\); roots.
RF13. Domain of \(\frac{x}{x^{2}-1}\).
RF14. \(\sum_{i=0}^{4}3^{i}\); \(\sum_{i=1}^{n}2\).
RF15. Inverse of \(f(x)=5x-2\); is \(x^{3}\) invertible on \(\mathbb{R}\)?

RF answers (11–15): \([-1,5]\); \((x-4)(x-5)\), roots \(4,5\); \(x\neq\pm 1\); \(\frac{3^{5}-1}{2}=121\), \(2n\); \(\frac{x+2}{5}\), yes with cube root.

Timed second-pass strategy

1. One full \(2\times 2\) solved two ways + \(\det\) check.
   
2. One rational simplify with domain line.
   
3. One \(\sum\) induction or geometric proof.
   
4. One inequality with a negative multiply (mark the flip).
   
5. Update repair table by day number (11–21), not by vague “algebra.”

Tomorrow

Day 23 — Propositions & truth tables. Stage III begins: what is a proposition; connectives; full truth tables; tautology and contradiction.