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Sum of a Finite Geometric Sequence

Sum of a Finite Geometric Sequence

Description:

This lesson explains how to the find the sum of a finite geometric sequence.

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Tutorial

  • Formula for the Sum of a Geometric Sequence
  • Geometric Sum when r > 1
  • Geometric Sum when r < 0
  • Partial Sums

Formula for the Sum of a Geometric Sequence

Consider the following geometric sequence: left curly bracket 24 comma space 12 comma space 6 comma space 3 right curly bracket

To find the sum of this finite sequence, it is simple enough to concretely add together all of the terms: 24 + 12 + 6 + 3 = 45.  However, there is a useful formula we can use in cases where this may be difficult to do (if there were 100 terms, for example). 


S subscript n equals a subscript 1 • left parenthesis fraction numerator 1 minus r to the power of n over denominator 1 minus r end fraction right parenthesis

where:

  • S subscript n is the sum of n terms
  • a subscript 1 is the value of the first term
  • r is the common ratio
  • n is the number of terms

We can confirm that the sum of the four terms above is 45 by using the formula as well:


Geometric Sums when r > 1

Our formula works for any value of r, although when we are working through the calculations, it may seem as though something must be off.  Don't worry, as long as you follow the steps properly, you should arrive at the correct sum.

Consider this sequence:

left curly bracket space 32 comma space 48 comma space 72 comma space 108 comma space 162 space right curly bracket

In this sequence, the value of the first term is 32, the common ratio is 1.5 (found by dividing any two consecutive terms, such as 108 ÷ 72 = 1.5), and n = 5, since we have 5 terms in the sequence. Applying the formula, we can find the sum of these 5 terms:


Checking concretely: 32 + 48 + 72 + 108 + 162 = 422

Geometric Sums when r < 0

The formula even works if the common ratio is a negative number.  Consider the formula: left curly bracket 6 comma space minus 18 comma space 54 comma space minus 162 comma space 486 right curly bracket

Here, the common ratio is –3, because we multiply 6 by –3 to get –18, and –18 by –3 to get 54, and so on.  To find the sum of these 5 terms using the formula, we take the following steps:


Confirming concretely: 6 – 18 + 54 – 162 + 486 = 366

Partial Sums

We can even use the formula to find a partial sum of a geometric sequence.  A partial sum means that we add some of the terms in the sequence, but not all of them.  Consider this sequence with 12 terms:

left curly bracket thin space 13 comma space 26 comma space 52 comma space 104 comma space 208 comma space 416 comma space 832 comma space 1664 comma space 3328 comma space 6656 comma space 13312 comma space 26624 space right curly bracket

If we want to find the sum of terms 3 through 8, we would make the following adjustments to our formula:

  • We will consider a1 to be 52. Although it is the 3rd term in the sequence, it is the 1st term we wish to include in the sum.
  • We will use n = 6 because the sum of terms 3 through 8 consists of 6 terms altogether

Everything else about the formula remains the same:


Adding concretely: 52 + 104 + 208 + 416 + 832 + 1664 = 3276