Geometric Sequences: nth Term, Sum and Consecutive Terms
A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant ratio r: an = a1 rn−1. In brief: the sequence 3, 6, 12, 24, … has a1 = 3 and r = 2, so an = 3 · 2n−1. The sum of the first n terms is Sn = a1(rn − 1)/(r − 1) for r ≠ 1, so S5 = 3(32 − 1)/1 = 93. Each term is the geometric mean of its neighbours: an2 = an−1an+1.
Geometric sequences describe repeated multiplication: compound interest, population growth, radioactive decay and the bounce heights of a ball. They are the discrete analogue of exponential functions. This guide explains the definition, the nth term, the sum, the property of consecutive terms and how to recognise a geometric sequence, with a table of examples.
What is a geometric sequence?
A sequence (an) with nonzero terms is geometric if the ratio r = an+1/an is the same for all n. Examples: 2, 6, 18, 54, … (r = 3); 80, 40, 20, 10, … (r = 1/2); 1, −2, 4, −8, … (r = −2). If |r| > 1 the terms grow in magnitude, if |r| < 1 they shrink towards 0, if r < 0 the signs alternate, and if r = 1 the sequence is constant. The behaviour matches the exponential functions, with the base r.
What is the formula for the nth term?
Starting from a1 and multiplying by r each time: an = a1 rn−1. More generally an = ak rn−k.
Example 1
a1 = 5, r = 3. a6 = 5 · 35 = 1,215.
Example 2: from two terms
a2 = 12 and a5 = 324. Then r3 = 324/12 = 27, so r = 3, and a1 = 4. The sequence is 4, 12, 36, 108, 324, …
Example 3: how many terms?
How many terms does 2, 6, 18, …, 4374 have? 2 · 3n−1 = 4374 gives 3n−1 = 2187 = 37, so n = 8.
What is the formula for the sum?
For r ≠ 1: Sn = a1(1 − rn)/(1 − r) = a1(rn − 1)/(r − 1). For r = 1: Sn = n a1.
Derivation: Sn = a1 + a1r + … + a1rn−1. Multiply by r: rSn = a1r + … + a1rn. Subtract: Sn − rSn = a1 − a1rn, so Sn(1 − r) = a1(1 − rn). An equivalent form with the last term is Sn = (anr − a1)/(r − 1).
Example 4
1 + 2 + 4 + … + 29 (10 terms): S = (210 − 1)/(2 − 1) = 1,023.
Example 5
Sum of 81, 27, 9, … for 6 terms: r = 1/3 and S6 = 81(1 − (1/3)6)/(1 − 1/3) = 81 · (728/729) · (3/2) = 364/3 ≈ 121.33. Direct check: 81 + 27 + 9 + 3 + 1 + 1/3 = 121.33 ✓.
What is the property of consecutive terms?
In a geometric sequence, the square of any term (except the first) equals the product of its neighbours: an2 = an−1 · an+1, so |an| is the geometric mean of the neighbours. Three nonzero numbers a, b, c are consecutive terms of a geometric sequence if and only if b2 = ac.
Example 6
For which x are x + 1, 2x + 2 and 5x + 1 consecutive terms of a geometric sequence? (2x + 2)2 = (x + 1)(5x + 1), so 4(x + 1)2 = (x + 1)(5x + 1). Either x + 1 = 0, which gives zero terms (not allowed), or 4(x + 1) = 5x + 1, so x = 3. The terms are 4, 8, 16 with r = 2.
Example 7: inserting geometric means
Insert two numbers between 5 and 135 so that all four are geometric. 135 = 5r3, so r = 3, and the numbers are 15 and 45.
Example 8: three numbers
Three numbers in a geometric sequence have the sum 21 and the product 216. Let them be b/r, b, br. The product gives b3 = 216, so b = 6. The sum 6/r + 6 + 6r = 21 gives 6r2 − 15r + 6 = 0, so 2r2 − 5r + 2 = 0 and r = 2 or 1/2. The numbers are 3, 6, 12 (or 12, 6, 3).
Worked examples
Example 9: a bouncing ball
A ball dropped from 10 m bounces to 60% of its previous height each time. After the 5th bounce the height is 10 · 0.65 = 0.7776 m. The heights are a geometric sequence with r = 0.6.
Example 10: compound interest
1,000 dollars at 5% compounded yearly: the balance after n years is 1000 · 1.05n. After 10 years it is 1,628.89 dollars. The sequence is geometric with r = 1.05.
Example 11: a folding paper
A sheet of paper 0.1 mm thick folded n times has the thickness 0.1 · 2n mm. After 10 folds: 102.4 mm. After 20 folds: 104,857.6 mm ≈ 105 m, although in practice, a sheet cannot be folded that many times.
Table of geometric sequences
| First terms | a1 | r | an | S6 |
|---|---|---|---|---|
| 1, 2, 4, 8, … | 1 | 2 | 2n−1 | 63 |
| 3, 9, 27, 81, … | 3 | 3 | 3n | 1,092 |
| 64, 32, 16, 8, … | 64 | 1/2 | 64 · (1/2)n−1 | 126 |
| 1, −2, 4, −8, … | 1 | −2 | (−2)n−1 | −21 |
| 5, 5, 5, 5, … | 5 | 1 | 5 | 30 |
| 100, 110, 121, … | 100 | 1.1 | 100 · 1.1n−1 | 771.56 |
Where does the sequence come from?
Geometric progressions appear in the Rhind papyrus (a problem with houses, cats and mice, sometimes read as the powers of 7), in Babylonian tablets and in Euclid’s Elements (Book IX, Proposition 35, gives the sum of a geometric progression). The legend of the grains of wheat on a chessboard, doubling on each square, illustrates the rapid growth: the total 264 − 1 ≈ 1.8 × 1019 grains. The legend exists in several versions, and its origin is uncertain.
Where is it used today?
Compound interest, depreciation at a fixed percentage, population models and the decay of radioactive substances are geometric. In computer science, a binary tree has 2k nodes at depth k, and the total with d levels is 2d+1 − 1. Musical pitches in equal temperament form a geometric sequence with the ratio 21/12 ≈ 1.0595 for a semitone.
Which mistakes are common?
Confusing rn with rn−1
an = a1rn−1. The first term has the exponent 0.
Mixing up ratio and difference
For 2, 6, 18 the ratio is 3 (not 4, which is the first difference).
Forgetting negative ratios
When solving r2 = 9, both r = 3 and r = −3 are possible, unless the terms are all positive.
Using the sum formula with r = 1
For r = 1 the formula divides by zero. The sum is n a1.