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Unit 3 · Sequences & Series
Geometric
Sequences
& Series
From foundations to infinite sums — exam-ready in one session
20 Problems
SAT / IB / AP Level
40 min Timer
Full Solutions
Core Concepts
What You Need to Know
Definition
Geometric Sequence
A sequence where each term is obtained by multiplying the previous term by a fixed constant called the common ratio \(r\).
\(a_1,\; a_1 r,\; a_1 r^2,\; a_1 r^3,\; \ldots\)
The common ratio: \(\displaystyle r = \frac{a_{n+1}}{a_n}\) for any \(n \geq 1\).
Formula 1
n-th Term
\(a_n = a_1 \cdot r^{\,n-1}\)
where \(a_1\) = first term, \(r\) = common ratio, \(n\) = term number.
Formula 2
Sum of First n Terms
\(S_n = \dfrac{a_1(r^n - 1)}{r - 1} \quad (r \neq 1)\)
Equivalently: \(\displaystyle S_n = \frac{a_1(1 - r^n)}{1 - r}\). When \(r = 1\), \(S_n = n\cdot a_1\).
Formula 3
Infinite Geometric Series
When \(|r| < 1\), the series converges:
\(S_\infty = \dfrac{a_1}{1 - r}\)
When \(|r| \geq 1\), the series diverges (no finite sum).
Formula 4
Geometric Mean
If \(b\) is the geometric mean of \(a\) and \(c\), then:
\(b^2 = ac \quad \Longrightarrow \quad b = \sqrt{ac}\)
Three terms \(a, b, c\) form a geometric sequence if and only if \(b^2 = ac\).
★ Must Memorize
01
n-th term: \(a_n = a_1 r^{n-1}\). The exponent is \((n-1)\), not \(n\). Don't confuse with \(a_1 r^n\).
02
Finite sum: \(S_n = \frac{a_1(r^n-1)}{r-1}\). Memorize both forms; use whichever avoids negative denominators.
03
Infinite sum: \(S_\infty = \frac{a_1}{1-r}\) ONLY when \(|r|<1\). Always check the condition first!
04
Finding r: If \(a_m\) and \(a_n\) are known, then \(r^{n-m} = \frac{a_n}{a_m}\).
05
Geometric mean test: Three terms form a geometric sequence \(\Leftrightarrow\) middle term squared equals product of outer terms.
06
Repeating decimals as infinite series: e.g. \(0.\overline{3} = \frac{3/10}{1-1/10} = \frac{1}{3}\).
Worked Examples
Try Before You Quiz
Example 1 — n-th Term
Find the 8th term of the geometric sequence 2, 6, 18, …
Identify: \(a_1 = 2\), \(r = 3\).
Apply formula: \(a_8 = 2 \cdot 3^{8-1} = 2 \cdot 3^7 = 2 \cdot 2187 = 4374\).
Answer: 4374
Example 2 — Finite Sum
Find \(S_5\) for the sequence 1, 2, 4, 8, …
\(a_1 = 1,\; r = 2,\; n = 5\).
\(S_5 = \frac{1(2^5-1)}{2-1} = \frac{32-1}{1} = 31\).
Answer: 31
Example 3 — Infinite Series
Find the sum of \(8 + 4 + 2 + 1 + \cdots\)
\(a_1 = 8,\; r = \frac{1}{2}\). Since \(|r| = \frac{1}{2} < 1\), the series converges.
\(S_\infty = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 16\).
Answer: 16
Example 4 — Finding a1 from S∞
An infinite geometric series has \(S_\infty = 20\) and \(r = \frac{3}{4}\). Find \(a_1\).
\(S_\infty = \frac{a_1}{1-r} = 20 \;\Rightarrow\; a_1 = 20\left(1 - \tfrac{3}{4}\right) = 20 \cdot \tfrac{1}{4} = 5\).
Answer: a₁ = 5
QUIZ COMPLETE
Answer Key & Solutions
Complete worked solutions for all 20 problems