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All courses Algebra II · A-SSE.4 24 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 \(4~+~12~+~36~+~108~+~324\), state the first term, verify the common ratio with consecutive quotients, and count the terms.
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Show answer choicesHide answer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
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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 and count the displayed terms

\[4,12,36,108,324~->~\text{first}~\text{term}~4;~\text{term}~\text{count}~5\]

The first addend is 4, and counting every listed addend gives five terms.

02

Verify a constant ratio

\[12/4=3;~36/12=3;~108/36=3;~324/108=3\]

Every consecutive quotient equals 3, so the sequence is geometric with positive common ratio 3.

03

State the series data

\[\text{First}~\text{term}:~4.~\text{Quotient}~\text{check}:~12/4=36/12=3.~\text{Common}~\text{ratio}:~3.~\text{Term}~\text{count}:~5.\]

The verified quotient and complete term count give the requested classification, choice A.

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Another way

  1. Observe that each term is the preceding term multiplied by 3, then confirm with at least two quotients.

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Common mistake

Count all five displayed addends; the number of multiplication steps between them is only four.

Solution walkthrough video