Calculus How To

Finite Geometric Sequence

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Sequence and Series >

A finite geometric sequence is a list of numbers (terms) with an ending; each term is multiplied by the same amount (called a common ratio) to get the next term in the sequence.
For example: the sequence 5, 10, 20, 40, 80, … 320 ends at 320. Each term is multiplied by 2 to get the next term.

Note: A slightly different form is the geometric series, where terms are added instead of listed: a + ar + ar2 + ar3 + …. These behave differently, and their sums are different. This article is about the geometric sequence; If you want to learn about the series, see: What is a Geometric Series?

Nth Term of a Geometric Sequence

The general form of the sequence is a1,  a1r2,   a1r3,   a1r4,…  a1r(n – 1)
.We don’t always want to write out the entire sequence all the time, so instead of writing everything out (5, 10, 20, 40, 80, 160…) we can use a much shorter formula. The general formula for the nth term of a geometric sequence is:

an = a1r(n – 1)

Where:

  • a1 = the first term in the sequence,
  • r = the common ratio.
  • n = the nth term.

For the example sequence above, the common ratio is 2 and the first term is 5. We can find out the nth terms by plugging those into the formula:
an = 5 · 2(n – 1).

  • First term: 5 · 2(1 – 1) = 5 · 2(0) = 5 · 1 = 5
  • Second term: 5 · 2(2 – 1) = 5 · 2(1) = 5 · 2 = 10
  • Third term: 5 · 2(3 – 1) = 5 · 2(2) = 5 · 4 = 20
  • Fourth term: 5 · 2(4 – 1) = 5 · 2(3) = 5 · 8 = 40

Another example: let’s say you are given 6(3)n – 1 and you’re asked to find the first five terms. Note: The first term in the formula is always in the first position, you know the first term here is 6.

  • First term: 6
  • Second term: 6 · 32 – 1 = 6 · 31 = 18
  • Third term: 6 · 33 – 1 = 6 · 3 2 = 54
  • Fourth term: 6 · 34 – 1 = 6 · 3 3 = 162
  • Fifth term: 6 · 35 – 1 = 6 · 3 4 = 486

Sum of a Finite Geometric Sequence

The sum of a finite geometric sequence is given by the formula (Larson, 2014):
sum of geometric sequence
Where:

Example question: What is the sum of the first 7 terms of a finite geometric series if the first term (a1 = 1 and the common ratio (r) = 2?
example of sum of finite geometric sequence


References

Harrison, B. (2020). Take Your Medicine. Adapted from Section 9.1 in Hughes-Hawlett, Deborah, et.al; Single Variable Calculus; John Wiley & Sons, Inc.; New York; 2002
Larson, R. (2014). College Algebra. Cengage Learning.
Seward, K. (2011). College Algebra: Tutorial 54D: Geometric Sequences and Series. Retrieved August 24, 2020 from: https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut54d_geom.htm

CITE THIS AS:
Stephanie Glen. "Finite Geometric Sequence" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/finite-geometric-sequence/
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