Arithmetic and Geometric Series

Finding sums of sequences

Series and Summation

What is a Series?

A series is the sum of the terms in a sequence.

Example: $2 + 4 + 6 + 8 + 10 = 30$

Arithmetic Series

Sum of an arithmetic sequence: $S_n = \frac{n(a_1 + a_n)}{2}$

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

where $n$ = number of terms

Geometric Series

Sum of a geometric sequence: $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$

(when $r \neq 1$ )

Infinite Geometric Series

If $|r| < 1$ , the infinite series has a sum: $S = \frac{a_1}{1 - r}$

Example: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$

$S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2$

Sigma Notation

$\sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + ... + a_n$

Read as: "the sum from $i = 1$ to $n$ of $a_i$ "

📚 Practice Problems

1Problem 1easy

❓ Question:

Find the sum: $2 + 5 + 8 + 11 + 14$

💡 Show Solution

This is an arithmetic series with:

$a_1 = 2$
$a_n = 14$
$n = 5$ terms

Use the formula: $S_n = \frac{n(a_1 + a_n)}{2}$

$S_5 = \frac{5(2 + 14)}{2} = \frac{5(16)}{2} = \frac{80}{2} = 40$

Answer: $40$

2Problem 2medium

❓ Question:

Find the sum of the first 6 terms: $3, 6, 12, 24, ...$

💡 Show Solution

This is a geometric series with:

$a_1 = 3$
$r = 2$
$n = 6$

Use the formula: $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$

$S_6 = 3 \cdot \frac{1 - 2^6}{1 - 2}$ $= 3 \cdot \frac{1 - 64}{-1}$ $= 3 \cdot \frac{-63}{-1}$ $= 3 \cdot 63 = 189$

Answer: $189$

3Problem 3hard

❓ Question:

Find the sum of the infinite series: $8 + 4 + 2 + 1 + ...$

💡 Show Solution

This is an infinite geometric series with:

$a_1 = 8$
$r = \frac{4}{8} = \frac{1}{2}$

Since $|r| = \frac{1}{2} < 1$ , the series converges.

Use the formula: $S = \frac{a_1}{1 - r}$

$S = \frac{8}{1 - \frac{1}{2}}$ $= \frac{8}{\frac{1}{2}}$ $= 16$

Answer: $16$

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