Arithmetic and Geometric Series

Number patterns and
sigma notation

Review

Geometric Sequence

A sequence is geometric when each term
after the first is found by multiplying by a non-
zero number, called the common ratio (r).

The common ratio can be found by taking any
term and dividing by the previous term:

𝑟 = 𝑢𝑛

𝑢𝑛−1

When r< -1 or r> 1, the sequence is diverging.
When -1 < r < -1, r ≠ 0, the sequence is
converging.

A sequence is called diverging if the absolute
value of each subsequent term is getting
larger, tending away from 0. A sequence is
called converging if the absolute value of
each subsequent term is getting smaller,
approaching 0.

Example

For each geometric progression, write down
the value of r and find the term indicated.

Practice:

Arithmetic and geometric series

Arithmetic series

The sum of the terms of an arithmetic sequence
is called an arithmetic series.

Since we already have the formula 𝑢𝑛 =
𝑢1 𝑛 − 1 𝑑, we can substitute it into the sum
formula:

Geometric series

Example:

For each infinite series below, decide whether it is
arithmetic or geometric, then find the sum to the
given term:

a.

b.

1+5+9+…𝑢8

−2 − 4 − 8 − ⋯ 𝑢9

​

​

Multiple Choice

Evaluate S_n:

28 + 35 + 42 + 49 + ..., n = 10

(Hint: is this arithmetic or geometric?)

• 2384

• 595

• 1192

• 1190

Multiple Choice

What is the sum of the first 50 terms of the series 2 + 17 + 32 + 47 + ...?

• 1,600
• 18,235
• 18,475
• 19,125

Multiple Choice

Evaluate the sum of the geometric series:

-1 + 5 - 25 + 125 + ... + 78,125

• 65104

• 56732

• -1/6

• 65816

Multiple Choice

Evaluate the geometric series below:
-4 - 8 - 16 - 32...,  S₆

• -253
• -252
• 4
• -235