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Algebra 2 Sequences and series

Finite geometric series formula

20 practice questions 0 video lessons Theory + worked examples

Finite Geometric Series Formula

Common Core Algebra 2 • Standard A-SSE.4 • Sequences & Series

Finite Geometric Series Formula is a topic in Sequences & Series in the Common Core State Standards. It is aligned to Standard A-SSE.4, which requires students to derive the formula for the sum of a finite geometric series and use it to solve problems.

The finite geometric series formula \(S_n=\dfrac{a_1(1-r^n)}{1-r}\) sums \(n\) terms, derived by multiplying by \(r\) and subtracting.

Common Core Algebra 2 › Sequences & Series › Finite Geometric Series Formula  —  Standard A-SSE.4

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Theory

The finite geometric series sum has a closed formula:

\[S_n=\dfrac{a_1(1-r^n)}{1-r},\qquad r\neq1.\]

It comes from multiplying \(S_n\) by \(r\) and subtracting, which cancels the middle terms.

If \(r=1\) the formula breaks (division by \(0\)); the sum is just \(n\,a_1\).
Finite geometric sum Finite geometric sum Finite geometric sum Sₙ = a₁(1 - rⁿ) / (1 - r), r ≠ 1 a₁ = first term, r = ratio n = number of terms
The finite geometric sum formula.
Why it works Why it works Why it works Sₙ = a₁ + a₁r + ... + a₁rⁿ⁻¹ r·Sₙ = a₁r + ... + a₁rⁿ subtract: Sₙ(1 - r) = a₁(1 - rⁿ)
The multiply-and-subtract derivation.

The sum of \(n\) terms:

\[S_n=\dfrac{a_1(1-r^n)}{1-r}\]
the sum is the first term times one minus r to the n over one minus r
Watch the signs when \(r>1\): both parts are negative.

How to use the formula

  1. Identify \(a_1\), \(r\), and \(n\).
  2. Compute \(r^n\).
  3. Substitute into \(S_n=\dfrac{a_1(1-r^n)}{1-r}\).
  4. Simplify carefully with signs.
Example 1 — Apply the formula
Find \(3+6+12+24+48\).
Solution

Geometric: \(a_1=3,\ r=2,\ n=5\).

\(S_5\)\(=\)\(\dfrac{3(1-2^5)}{1-2}\)
\(=\)\(\dfrac{3(-31)}{-1}=93\)
the sum is 93
Example 2 — Ratio 3
Find \(1+3+9+27\).
Solution

\(a_1=1,\ r=3,\ n=4\).

\(S_4\)\(=\)\(\dfrac{1(1-3^4)}{1-3}\)
\(=\)\(\dfrac{-80}{-2}=40\)
the sum is 40
Example 3 — A fractional ratio
Find \(16+8+4+2\).
Solution

\(a_1=16,\ r=\dfrac12,\ n=4\).

\(S_4\)\(=\)\(\dfrac{16\left(1-\dfrac1{16}\right)}{1-\dfrac12}\)
\(=\)\(\dfrac{15}{\dfrac12}=30\)
the sum is 30
Example 4 — When r = 1
What is the sum if \(r=1\)?
Solution

Every term equals \(a_1\), so the formula is not needed.

\(S_n\)\(=\)\(n\cdot a_1\)
the sum is n times the first term

Common pitfalls

Count \(n\) correctly — it is the number of terms.
\(r=1\) needs \(S_n=n a_1\) instead of the formula.
Signs: for \(r>1\), numerator and denominator are both negative.

Frequently asked questions

What is the finite geometric series formula?

\(S_n=\dfrac{a_1(1-r^n)}{1-r}\).

How is the formula derived?

Multiply the sum by \(r\), subtract, and solve — the middle terms cancel.

What if \(r=1\)?

All terms are equal, so the sum is \(n\,a_1\).

What is \(n\) in the formula?

The number of terms being added.

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