Geometric Sequences and Series

Replace constant addition with constant multiplication and terms race away, collapse toward zero, or flip sign at every step — all decided by the common ratio.

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Replace the constant addition of an arithmetic sequence with a constant multiplication and the behaviour changes completely. Terms no longer creep along a straight line; they race away, collapse toward zero, or flip sign at every step — all decided by a single number, the common ratio.

Theorem The two core formulas

The th term is , and the sum of the first terms is , for .

Here is the first term, the common ratio, and the position.

Term formula:
Sum, knowing and :
Sum, knowing the last term:

Find by dividing any term by the one before it, just as you found by subtracting.

Concept What the ratio controls
— the terms grow
0 < r < 1 — the terms shrink toward zero
r < 0 — the terms alternate in sign

Each bar is double the one before it. After only a handful of steps the growth is dramatic — which is exactly why chain letters and compound interest are modelled this way.

Example Finding a term

a) An email chain

Ahmed sends emails in the first stage, and every recipient forwards it to more. How many emails are sent in the eighth stage?

, ,
Substitute:
Evaluate:
emails

b) Write a rule for

Common ratio:

c) Given and , find

Substitute:
Simplify:
Example Geometric means

Find three geometric means between and .

Three means plus the two end terms gives
Substitute:
Simplify:
Take the fourth root: or
If :
If :
The means are or

An even power hides a sign, so both answers are genuine. This never happens with arithmetic means, where solving for is a linear step with a single result.

Example Summing a series

a) Total emails through the eighth stage

, ,
Substitute:
Evaluate the power:
Simplify:
emails in total

b) Evaluate

First term at :
Number of terms:
Common ratio:
Substitute:

Counting the terms is where marks are lost. From to there are terms, not — subtract, then add one back.

Summary
  1. A geometric sequence multiplies by a constant ratio instead of adding a constant .
  2. Find by dividing any term by the one before it.
  3. The th term is .
  4. The sum of terms is , valid whenever .
  5. grows, 0 < r < 1 shrinks, and r < 0 alternates in sign.
  6. Inserting geometric means makes terms, and an even root gives two valid answers.
  7. In sigma notation the number of terms is top bottom .
283. Relations and Functions
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