Sequences and Series · Lesson 2

Geometric sequences

A geometric sequence multiplies by the same number, the common ratio rr, to get from each term to the next. Find rr by dividing a term by the one before it: in 3, 6, 12, 24, ... the ratio is 63=2\frac{6}{3} = 2. A ratio between 0 and 1 makes the terms shrink, and a negative ratio makes the signs alternate.

As with arithmetic sequences, the nnth term is n−1n - 1 steps after the first, so an=a1rn−1a_n = a_1 r^{n-1}. The exponent goes on rr only: 3⋅273 \cdot 2^7 is not 676^7.

Given two terms kk steps apart, the later term is the earlier one times rkr^k. So divide the terms and take the kkth root: if a2=6a_2 = 6 and a5=−162a_5 = -162, then r3=−1626=−27r^3 = \frac{-162}{6} = -27 and r=−3r = -3. When kk is even, the root can be positive or negative (r2=4r^2 = 4 gives r=2r = 2 or r=−2r = -2), so use the other information in the question to decide.

To tell the two kinds apart, check differences and quotients. Equal differences mean arithmetic; equal quotients mean geometric. A list like 1, 4, 9, 16 has neither.

Arithmetic or geometric
SequenceCheckKind
3,6,9,12,…3, 6, 9, 12, \ldotsDifferences all 3Arithmetic, d=3d = 3
3,6,12,24,…3, 6, 12, 24, \ldotsQuotients all 2Geometric, r=2r = 2
80,−40,20,−10,…80, -40, 20, -10, \ldotsQuotients all −12-\frac{1}{2}Geometric, r=−12r = -\frac{1}{2}
1,4,9,16,…1, 4, 9, 16, \ldotsDifferences 3, 5, 7; quotients changeNeither

Worked example Easy

The first term of a geometric sequence is 5, and the common ratio is 2. What is the 7th term of the sequence?

  1. 1717
  2. 6060
  3. 160160
  4. 320320Answer

How to solve it

  1. The 7th term is 7−1=67 - 1 = 6 steps after the 1st, so multiply by 2 six times.
  2. a7=5⋅26=5⋅64=320a_7 = 5 \cdot 2^{6} = 5 \cdot 64 = 320.
  3. Check by listing: 5, 10, 20, 40, 80, 160, 320.

Why each choice is right or wrong

  • A. Incorrect. 17 is 5+6⋅25 + 6 \cdot 2, which adds the ratio at each step. In a geometric sequence you multiply by it.
  • B. Incorrect. 60 is 5⋅(2⋅6)5 \cdot (2 \cdot 6), which multiplies the ratio by the number of steps. Six steps means 262^6, not 2⋅62 \cdot 6.
  • C. Incorrect. 160 is 5⋅255 \cdot 2^5, the 6th term. One more step gives the 7th term.
  • D. Correct. a7=5⋅27−1=5⋅64=320a_7 = 5 \cdot 2^{7 - 1} = 5 \cdot 64 = 320.

Worked example Medium

In a geometric sequence, the 2nd term is 6 and the 5th term is −162-162. What is the 6th term of the sequence?

  1. −1,458-1{,}458
  2. −486-486
  3. 486486Answer
  4. 4,3744{,}374

How to solve it

  1. From the 2nd term to the 5th term is 3 steps, so r3=−1626=−27r^3 = \frac{-162}{6} = -27.
  2. The real cube root of −27-27 is −3-3, so r=−3r = -3.
  3. One more step: a6=−162⋅(−3)=486a_6 = -162 \cdot (-3) = 486.
  4. Check: the sequence is −2,6,−18,54,−162,486-2, 6, -18, 54, -162, 486; the signs alternate, as they must with a negative ratio.

Why each choice is right or wrong

  • A. Incorrect. −1,458-1{,}458 is −2⋅(−3)6-2 \cdot (-3)^6, which uses the exponent nn instead of n−1n - 1. That is the 7th term.
  • B. Incorrect. −486-486 multiplies −162-162 by 33, dropping the negative sign of the ratio. With r=−3r = -3 the sign flips each step.
  • C. Correct. r3=−1626=−27r^3 = \frac{-162}{6} = -27, so r=−3r = -3, and a6=−162⋅(−3)=486a_6 = -162 \cdot (-3) = 486.
  • D. Incorrect. 4,3744{,}374 multiplies −162-162 by −27-27, which is r3r^3, not rr. Take the cube root to get the ratio.

Trap Using $r^n$ instead of $r^{n-1}$

The 7th term is 6 multiplications after the first, so its power of rr is 6. A choice equal to a1rna_1 r^n is the next term in the list. If you are unsure, test the formula on the 2nd term: a2=a1r1a_2 = a_1 r^1.

Trap Adding the ratio

In a geometric sequence the ratio multiplies; it is never added. And a ratio found from terms 3 steps apart is r3r^3, not rr: take the cube root before you use it.

Calculator When a calculator is faster

A calculator makes large powers such as 3⋅273 \cdot 2^{7} or 5⋅355 \cdot 3^{5} quick and safe, and it takes cube roots like −273\sqrt[3]{-27}. In the built-in Desmos calculator, if you test online, type 3*2^(n-1) into a table to list terms. Finding the ratio and counting the steps is still your job.