Geometric Sequence — Formula, Sum, Examples
Geometric Sequence — Formula, Sum, Examples
TL;DR
A geometric sequence is a list of numbers where every term after the first is found by multiplying the previous term by a fixed common ratio r. This article covers the nth-term and sum formulas, three worked examples at Quick/Standard/Stretch tiers, the role of |r|<1 for convergent infinite series, and the difference between a sequence and a series.
A List Whose Next Term Is Always a Multiplication Away
Some lists of numbers grow by adding a fixed amount. Others grow by multiplying by a fixed amount. The second kind — the geometric sequence — is what compound interest looks like, what radioactive decay looks like, and what a viral video's view count looks like in its first 24 hours.
A geometric sequence is one of the smallest mathematical objects that captures the essence of exponential change. Master it once and the same pattern appears in twenty later topics.
What a Geometric Sequence Is
A geometric sequence is an ordered list of numbers a₁,a₂,a₃,… such that the ratio between any two consecutive terms is the same constant r:
[ \frac{a_{n+1}}{a_n} = r \quad \text{for every } n. ]
r is the common ratio, a₁ = a is the first term, and the general nth term is
[ a_n = a \cdot r^{n - 1}. ]
Examples — 5,10,20,40,… (a=5, r=2). ; 81,27,9,3,1,… (a=81, r=1/3). ; 1,−2,4,−8,16,… (a=1, r=−2).
Quick facts.
- nth term: (a_n = a \cdot r^{n-1})
- Common ratio: (r = \frac{a_{n+1}}{a_n}), the same for every consecutive pair.
- Finite sum: (S_n = \frac{a(1 - r^n)}{1 - r}) for (r \neq 1).
- Infinite sum: (S_{\infty} = \frac{a}{1 - r}) when |r| < 1 (the series converges).
- Sequence vs series: a sequence is a list of terms; a series is the sum of a sequence's terms.
- Grade introduced: CBSE Class 11 (sequences chapter); CCSS-M HSF-LE.A.2 (construct linear and exponential functions, including geometric sequences); NCERT Class 11 Chapter 9 — Sequences and Series.
The Sum of a Geometric Sequence
When the terms of a geometric sequence are added, the result is called a geometric series. The first n terms sum to
[ S_n = \frac{a(1 - r^n)}{1 - r}, \quad r \neq 1. ]
When r=1, every term equals a and (S_n = na).
When |r|<1 and we sum infinitely many terms, the r^n factor shrinks to zero, and the infinite sum becomes
[ S_{\infty} = \frac{a}{1 - r}. ]
When |r|≥1, the terms do not shrink and the infinite sum diverges (does not exist as a finite number).
Examples of Geometric Sequence
Quick. Find the 8th term of the sequence 2,4,8,16,…
Final answer: (a_8 = 256).
Standard (Wrong Path First — Watch How This Goes Wrong). Find the sum 1+1/2+1/4+1/8+⋯
Final answer: the infinite sum is exactly 2.
Stretch. The third term of a geometric sequence is 18 and the sixth term is 486. Find the first term and the common ratio. Final answer: a=2, r=3. The sequence is 2,6,18,54,162,486,…
Where Geometric Sequences Show Up
The geometric sequence is one of the most common patterns in nature, finance, and engineering.
- Compound interest. A principal P at rate r% compounded annually produces the geometric sequence P,P(1+r),P(1+r)²,…
- Radioactive decay. After each half-life, the remaining quantity is half the previous — geometric with r=1/2.
- Bouncing ball. A ball that bounces to 80% of its previous height produces a geometric sequence of peak heights with r=0.8.
- Repeating decimals. 0.333…=0.3+0.03+0.003+⋯ is a geometric series summing to 1/3.
- Population growth. Before resource limits hit, populations grow geometrically (the early phase of exponential growth).
The Geometric Sequence Errors That Cost Most Marks
- Using the arithmetic-sum formula on a geometric sequence. Use (S_n = a(1 - r^n)/(1 - r)).
- Assuming all infinite sums diverge. Check |r|. If |r|<1, the infinite sum converges.
- Confusing the first term with the second term. Use the first term as a.
- Forgetting the −1 in the exponent. Use (a_n = a \cdot r^{n-1}).
The Mathematicians Who Shaped Sequences
Pingala (c. 200 BCE, India) worked with geometric and arithmetic sequences in his Chandaḥśāstra. Archimedes (c. 287–212 BCE, Greece) computed the infinite sum 1/4+1/16+1/64+⋯=1/3. Fibonacci (c. 1170–1250, Italy) introduced sequences to medieval Europe in Liber Abaci (1202).
Conclusion
- A geometric sequence has a constant ratio r between consecutive terms.
- The nth term is (a_n = a r^{n-1}).
- The finite sum is (S_n = a(1 - r^n)/(1 - r)); the infinite sum is (a/(1 - r)) when |r| < 1.
- The most common mistake is using the arithmetic-sum formula on a geometric sequence.
- Geometric sequences model compound interest, radioactive decay, and the early phase of any exponential growth.
Try It Yourself — Three Problems
- Find the 10th term of 5,15,45,135.…
- Find the sum of the first 6 terms of 4,8,16,32.…
- Compute 1+1/3+1/9+1/27+⋯ (infinitely many terms).
Frequently Asked Questions
What is the difference between a geometric sequence and an arithmetic sequence?
An arithmetic sequence has a constant difference between terms. A geometric sequence has a constant ratio between terms.
What is the common ratio?
The fixed number r that multiplies one term to produce the next.
When does the infinite sum exist?
When |r|<1.
Can the common ratio be negative?
Yes. r=−2 gives 1,−2,4,−8,….
Can r=0 work?
No.
What is a geometric series?
The sum of the terms of a geometric sequence.