Geometric sequences

About this lesson

Precalculus › Sequences and series

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Geometric sequences

Warm-up

Three quick questions from earlier lessons.

Discover

A rumor spreads: 3 people know it on day 1, and each day the number triples. How many know it on day 4?

  1. 81
  2. 12
  3. 243

Key idea

A geometric sequence multiplies by the same number rr (the common ratio) each time.

an=a1⋅r n−1a_n = a_1 \cdot r^{\,n-1}, and r=an+1anr = \dfrac{a_{n+1}}{a_n}.

Worked example

Find a7a_7 for 2,6,18,…2, 6, 18, \dots

r=3r = 3
6 ÷ 2.
a7=2⋅36a_7 = 2 \cdot 3^6
6 multiplications from a₁ to a₇.
a7=1458a_7 = 1458
3⁶ = 729.

Your turn, guided

Your turn: a1=5a_1 = 5, r=2r = 2. Find a6a_6.

a6=5⋅25a_6 = 5 \cdot 2^5
5 multiplications.
a6=160a_6 = 160
5 · 32.

Practice ★

  1. A geometric sequence has a1=3a_1 = 3 and ratio 3. Find a4a_{ 4 }.
  2. A geometric sequence has a1=1a_1 = 1 and ratio 2. Find a8a_{ 8 }.
  3. A geometric sequence has a1=2a_1 = 2 and ratio 3. Find a8a_{ 8 }.
  4. A geometric sequence has a1=4a_1 = 4 and ratio 2. Find a5a_{ 5 }.

Watch out!

Practice ★★

  1. Find the common ratio of the geometric sequence 8,32,128,…8, 32, 128, \dots
  2. Find the common ratio of the geometric sequence 7,21,63,…7, 21, 63, \dots
  1. Write a formula for ana_n, starting at n=1n = 1: 2,8,32,…2, 8, 32, \dots
  2. Write a formula for ana_n, starting at n=1n = 1: 9,36,144,…9, 36, 144, \dots
  3. Write a formula for ana_n, starting at n=1n = 1: 8,40,200,…8, 40, 200, \dots

Exit ticket

Three questions to check you have got it.

  1. Write a formula for ana_n, starting at n=1n = 1: 2,8,32,…2, 8, 32, \dots
  2. Write a formula for ana_n, starting at n=1n = 1: 9,36,144,…9, 36, 144, \dots
  3. Write a formula for ana_n, starting at n=1n = 1: 8,40,200,…8, 40, 200, \dots

Summary card

Geometric sequences

  • Multiply by rr each time: an=a1r n−1a_n = a_1 r^{\,n-1}.
  • rr = a term divided by the one before.

Watch out: the exponent is n−1n - 1.

End of the lesson

Next lesson: Sigma notation and arithmetic series Back to the chapter

Answers: Geometric sequences

A

  1. 8181
  2. 128128
  3. 43744374
  4. 6464

B

  1. 44
  2. 33

C

  1. 2⋅4n−12 \cdot 4^{n - 1}
  2. 9⋅4n−19 \cdot 4^{n - 1}
  3. 8⋅5n−18 \cdot 5^{n - 1}

D

  1. 2⋅4n−12 \cdot 4^{n - 1}
  2. 9⋅4n−19 \cdot 4^{n - 1}
  3. 8⋅5n−18 \cdot 5^{n - 1}

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