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All courses Math III · A-SSE.4 17 of 55
Derive and use the finite geometric series formula to solve problems such as mortgage-payment models.

Identify and verify a finite geometric series

Problem
For the finite geometric series 4+12+36+108+324, enter the first term a, an adjacent-term quotient check, the common ratio r with its sign, and the integer term count n.
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Hint

Copy the first addend, divide the second by the first, and verify with the next pair.

For a geometric series, r=aₖ₊₁/aₖ is constant; a negative r alternates signs.

Solution walkthrough

01

Identify the first term

\[\text{first}~\text{term}~=~4\]

The first term of the series is simply the first number listed.

02

Find the common ratio

\[12/4~=~3\]

Dividing a term by the previous term shows the constant multiplier between terms.

03

Count the terms

\[4,~12,~36,~108,~324~->~5~\text{terms}\]

There are five numbers in the finite geometric series.

04

State all three quantities

\[\text{common}~\text{ratio}:~3;~\text{first}~\text{term}:~4;~\text{term}~\text{count}:~5\]

So the series starts at \(4\), multiplies by \(3\) each time, and contains \(5\) terms.

05

State the final answer

\[\text{first}~\text{term}~a=4;~\text{adjacent}-\text{ratio}~\text{check}=12/4=36/12=3;~\text{common}~\text{ratio}~r=3;~\text{integer}~\text{term}~\text{count}~n=5\]

Common ratio: \(3\); first term: \(4\); term count: \(5\).

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Another way

  1. You can check the ratio again with another pair, such as \(36/12=3\), to confirm it stays constant.

!

Common mistake

A common mistake is to subtract consecutive terms instead of dividing them, but geometric series are identified by a constant ratio, not a constant difference.