Chapter Review

Sequences and Series

Arithmetic Sequence · Geometric Sequence · Harmonic and Special Series

Arithmetic Sequence Basics

An arithmetic sequence has a constant common difference $d = a_n - a_{n-1}$ between consecutive terms, producing linear growth when plotted against index.

Key Points

•
Common difference: $d = a_n - a_{n-1}$ (must be identical for ALL consecutive pairs)
•
If $d > 0$ the sequence increases; if $d < 0$ it decreases; if $d = 0$ all terms are equal
•
Non-consecutive shortcut: $d = \frac{a_p - a_m}{p - m}$
•
General term: $a_n = a_1 + (n-1)d$ — a linear function of $n$ with slope $d$
•
To check membership: $n = \frac{a_n - a_1}{d} + 1$ must be a positive integer

Formula

$$a_n = a_1 + (n-1)d$$

Arithmetic Series (Summation)

The sum of $n$ terms of an A.P. is a quadratic function of $n$, computed via Gauss's pairing method or the standard expansion.

Key Points

•
Gauss form (when $a_n$ is known): $S_n = \frac{n}{2}(a_1 + a_n)$
•
Standard form (when $a_n$ is unknown): $S_n = \frac{n}{2}[2a_1 + (n-1)d]$
•
$S_n$ is quadratic in $n$: $S_n = \frac{d}{2}n^2 + (a_1 - \frac{d}{2})n$
•
When $S_n$ is given as a value, the resulting equation in $n$ is quadratic — reject the negative root

Formula

$$S_n = \frac{n}{2}[2a_1 + (n-1)d]$$

Arithmetic Mean and Inserted AMs

The arithmetic mean $A = \frac{a+b}{2}$ is the midpoint of two numbers; inserting $n$ AMs between $a$ and $b$ creates an $(n+2)$-term A.P.

Key Points

•
Single AM: $A = \frac{a+b}{2}$, equidistant from both endpoints
•
Any term in an A.P. is the AM of its two neighbours: $a_n = \frac{a_{n-1} + a_{n+1}}{2}$
•
Inserting $n$ AMs: common difference $d = \frac{b-a}{n+1}$ (there are $n+1$ gaps)
•
Sum of $n$ inserted AMs equals $n \times \frac{a+b}{2}$

Formula

$$d = \frac{b - a}{n + 1}$$

Geometric Sequence Basics

A geometric sequence multiplies each term by a constant common ratio $r$, producing exponential growth ($|r|>1$) or decay ($|r|<1$).

Key Points

•
Common ratio: $r = a_n / a_{n-1}$ (no term can be zero)
•
If $r > 1$: growth; if $0 < r < 1$: decay; if $r < 0$: alternating signs
•
Explicit formula: $a_n = a_1 r^{n-1}$ — exponent is $n-1$, not $n$
•
Finding $r$ from two terms: $r^{m-k} = a_m / a_k$
•
Even powers of a negative $r$ are positive; odd powers are negative

Formula

$$a_n = a_1 r^{n-1}$$

Geometric Series and Infinite Sums

A finite geometric series sums via the ratio formula; when $|r| < 1$, the infinite series converges to a finite limit as $r^n \to 0$.

Key Points

•
Finite sum: $S_n = \frac{a_1(1 - r^n)}{1 - r}$ (use $\frac{a_1(r^n - 1)}{r - 1}$ when $|r| > 1$)
•
Undefined at $r = 1$: use $S_n = na_1$ directly
•
Infinite sum ($|r| < 1$): $S_\infty = \frac{a_1}{1 - r}$
•
If $|r| \ge 1$ the series diverges (sum is unbounded or oscillates)
•
Every repeating decimal is an infinite G.P. convertible to a fraction via $S_\infty$

Formula

$$S_\infty = \frac{a_1}{1 - r}, \quad |r| < 1$$

Geometric Mean and Inserted GMs

The geometric mean $G = \pm\sqrt{ab}$ is the multiplicative midpoint; inserting $n$ GMs creates an $(n+2)$-term G.P.

Key Points

•
Single GM: $G = \pm\sqrt{ab}$ (real GM exists only if $a$ and $b$ have the same sign)
•
$G = +\sqrt{ab}$ gives positive ratio; $G = -\sqrt{ab}$ gives negative ratio
•
Inserting $n$ GMs: $r = (b/a)^{1/(n+1)}$
•
Product of all $n$ GMs equals $(\sqrt{ab})^n$
•
AM ≥ GM inequality: $\frac{a+b}{2} \ge \sqrt{ab}$ for positive $a, b$

Formula

$$G = \pm\sqrt{ab}$$

Harmonic Progression and Harmonic Mean

A harmonic progression has reciprocals forming an A.P.; all HP problems are solved by converting to the reciprocal AP domain.

Key Points

•
HP general term: $H_n = \frac{1}{a + (n-1)d}$ where $a, d$ belong to the underlying AP
•
No term of an HP can be zero (reciprocal would be undefined)
•
No direct sum formula for HP — always convert to AP first
•
Harmonic Mean: $H = \frac{2ab}{a+b}$ (reciprocal of AM of reciprocals)
•
Inserting $n$ HMs: insert $n$ AMs between $1/a$ and $1/b$, then take reciprocals

Formula

$$H = \frac{2ab}{a+b}$$

AM–GM–HM Relationship

For two positive reals, the three Pythagorean means satisfy $A \ge G \ge H$ with equality only when $a = b$, and are linked by $G^2 = AH$.

Key Points

•
$A = \frac{a+b}{2}$, $G = \sqrt{ab}$, $H = \frac{2ab}{a+b}$
•
Identity: $G^2 = A \cdot H$ (so $A, G, H$ are themselves in G.P.)
•
Inequality: $A > G > H$ for distinct positive reals
•
If two of $A, G, H$ are known, $a$ and $b$ are roots of $x^2 - 2Ax + G^2 = 0$

Formula

$$G^2 = A \cdot H$$

Sigma Notation and Power Sums

Sigma notation $\sum$ compactly expresses sums; closed-form formulas exist for sums of first $n$ natural numbers, their squares, and their cubes.

Key Points

•
$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$
•
$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$
•
$\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2$ (square of the sum of $n$)
•
Linearity: $\sum(ca_k + b_k) = c\sum a_k + \sum b_k$
•
The index variable is a dummy — the letter used doesn't affect the result

Formula

$$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$

Telescoping Series

A telescoping series expresses each term as a difference $T_r = V_r - V_{r-1}$, causing intermediate terms to cancel so the sum collapses to $V_n - V_0$.

Key Points

•
Cancellation: $(V_1 - V_0) + (V_2 - V_1) + \cdots + (V_n - V_{n-1}) = V_n - V_0$
•
Partial fractions are the key tool: e.g., $\frac{1}{r(r+1)} = \frac{1}{r} - \frac{1}{r+1}$
•
For infinite telescoping sums, $S_\infty = \lim_{n \to \infty} V_n - V_0$
•
Example: $\sum_{r=1}^{\infty} \frac{1}{r(r+1)} = 1$

Formula

$$\sum_{r=1}^{n} (V_r - V_{r-1}) = V_n - V_0$$

Formulas

AP General Term

The nth term of an arithmetic progression.

Formula

$a_n = a_1 + (n-1)d$

AP Sum (Standard)

Sum of first n terms using first term and common difference.

Formula

$S_n = \frac{n}{2}[2a_1 + (n-1)d]$

GP General Term

The nth term of a geometric progression.

Formula

$a_n = a_1 r^{n-1}$

GP Finite Sum

Sum of first n terms of a geometric series.

Formula

$S_n = \frac{a_1(1 - r^n)}{1 - r}$

GP Infinite Sum

Sum to infinity when |r| < 1.

Formula

$S_\infty = \frac{a_1}{1 - r}$

Harmonic Mean

HM between two numbers a and b.

Formula

$H = \frac{2ab}{a+b}$

Means Identity

Links AM, GM, and HM for two numbers.

Formula

$G^2 = A \cdot H$

Sum of Squares

Closed-form sum of squares of first n naturals.

Formula

$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$