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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:

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).

where:

- is the sum of n terms
- is the value of the first term
- is the common ratio
- 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:

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:

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:

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

- We will consider a
_{1}_{ }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

Formulas to Know

- Sum of a Finite Geometric Sequence