Arithmetic and Geometric Sequences and Series

By the end of this lesson, you will consolidate arithmetic and geometric sequence/series knowledge with sigma notation, recursive definitions, and connections between series and limits.

Sigma Notation

$$\sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n$$

Properties of sigma:

$$\sum_{k=1}^{n}(a_k + b_k) = \sum a_k + \sum b_k, \qquad \sum_{k=1}^{n} ca_k = c\sum_{k=1}^{n} a_k$$

Common sums:

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

Arithmetic Sequence and Series (Quick Reference)

$a_n = a_1 + (n-1)d$; $S_n = \dfrac{n}{2}(a_1 + a_n)$

Sigma form: $\displaystyle\sum_{k=1}^{n}[a_1 + (k-1)d] = na_1 + d\cdot\frac{n(n-1)}{2}$

Geometric Sequence and Series (Quick Reference)

$a_n = a_1r^{n-1}$; $S_n = a_1\dfrac{1-r^n}{1-r}$ ($r \neq 1$)

Infinite sum: $S_\infty = \dfrac{a_1}{1-r}$ for $|r| < 1$

Recursive Definitions

A recursive formula defines each term using previous terms:

Arithmetic: $a_n = a_{n-1} + d$ with $a_1$ given.

Geometric: $a_n = r \cdot a_{n-1}$ with $a_1$ given.

Fibonacci: $F_1 = F_2 = 1$; $F_n = F_{n-1} + F_{n-2}$ — famous recursive sequence.

Example 1 — Evaluating Sigma Sums

$\displaystyle\sum_{k=3}^{7}(2k-1)$: terms are $5, 7, 9, 11, 13$ → sum = 45.

Using formulas: $\sum_{k=3}^{7}(2k-1) = 2\sum_{k=3}^{7}k - 5 = 2\left(\frac{7(8)}{2} - \frac{2(3)}{2}\right) - 5 = 2(28-3) - 5 = 45$ ✓

Example 2 — Writing in Sigma Notation

$3 + 6 + 12 + \cdots + 96$: geometric, $a_1 = 3$, $r = 2$. Find $n$: $3 \cdot 2^{n-1} = 96 \implies 2^{n-1} = 32 \implies n = 6$.

$$\sum_{k=1}^{6} 3 \cdot 2^{k-1}$$

Connection to Limits: Series vs. Sequence of Partial Sums

The sequence of partial sums is $S_1, S_2, S_3, \ldots$ where $S_n = \sum_{k=1}^n a_k$.

An infinite series $\sum_{k=1}^\infty a_k$ converges iff $\lim_{n \to \infty} S_n$ exists and is finite.

$\lim_{n \to \infty} S_n = \dfrac{a_1}{1-r}$ when $|r| < 1$.