Key takeaway: A geometric series is the sum of terms that share a constant ratio r, with finite total S_n = a(1 − rⁿ)/(1 − r), and an infinite total a/(1 − r) when |r| < 1.
Grade 11 – Algebra 2A-SSE.B.4Expressions and Equations
Definition of Geometric Series
A geometric series adds up terms in which each one is the previous term multiplied by the same number, the common ratio \(r\). Doubling amounts, shrinking bounces and repeated percentage changes all create this pattern.
The sum has a closed form because multiplying the series by \(r\) shifts every term one place over, and subtracting the two lines cancels almost everything. When \(|r|\lt 1\) the terms fade toward zero, so even an endless geometric series can settle on a finite total. When \(|r|\ge 1\) the endless version has no finite sum.
Formula and notation
For first term \(a\), ratio \(r\neq 1\) and \(n\) terms:
\[S_n=\frac{a\,(1-r^{n})}{1-r}, \qquad S_{\infty}=\frac{a}{1-r}\ \text{ if } |r|\lt 1.\]
The ratio is found by dividing any term by the term before it.
Key facts
- The ratio r is the next term divided by the previous term, never the difference.
- If r = 1, the sum is simply n times the first term.
- An infinite sum exists only when the ratio lies strictly between -1 and 1.
- Sigma notation for a geometric series shows the ratio raised to growing powers.
Geometric Series example
A ball rebound sequence starts at 96 inches and each bounce is half the height of the one before. Find the total of the first 6 heights.
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Identify \(a=96\), \(r=\frac{1}{2}\) and \(n=6\).
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Substitute: \(S_6=\frac{96(1-(\frac{1}{2})^6)}{1-\frac{1}{2}}\).
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Simplify: \(\frac{96\cdot\frac{63}{64}}{\frac{1}{2}}=192\cdot\frac{63}{64}=189\).
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Check by adding: \(96+48+24+12+6+3=189\).
The six heights add up to 189 inches.
Common mistake
Watch out
Ibrahim looks at 96, 48, 24 and decides the ratio is \(\frac{96}{48}=2\), then computes \(S_6=\frac{96(1-2^6)}{1-2}=6048\).
Why it is wrong
The ratio is the later term over the earlier one: \(\frac{48}{96}=\frac{1}{2}\). Dividing the earlier term by the later one flips the ratio, and this geometric series then seems to total far more than the real 189.
Related terms and lessons
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