Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series.
formula for the sum of the first [latex]n[/latex] terms of a geometric series
A geometric series is the sum of the terms in a geometric sequence.
The formula for the sum of the first [latex]n[/latex] terms of a geometric sequence is represented as
Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio, [latex]r[/latex]. We can write the sum of the first [latex]n[/latex] terms of a geometric series as
Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first [latex]n[/latex] terms of a geometric series. We will begin by multiplying both sides of the equation by [latex]r[/latex].
Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for [latex]{S}_{n}[/latex], factor [latex]a_1[/latex] on the right hand side and divide both sides by [latex]\left(1-r\right)[/latex].
How To: Given a geometric series, find the sum of the first [latex]n[/latex] terms.
Identify [latex]{a}_{1},r,\text{ and }n[/latex].
Substitute values for [latex]{a}_{1},r[/latex], and [latex]n[/latex] into the formula [latex]{S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}[/latex].
Simplify to find [latex]{S}_{n}[/latex].
Use the formula to find the indicated partial sum of each geometric series.
[latex]{S}_{11}[/latex] for the series [latex]8 + -4 + 2 + \dots[/latex]
[latex]{a}_{1}=8[/latex], and we are given that [latex]n=11[/latex]. We can find [latex]r[/latex] by dividing the second term of the series by the first.
[latex]r=\dfrac{-4}{8}=-\frac{1}{2}[/latex]
Substitute values for [latex]{a}_{1}, r, \text{ and } n[/latex] into the formula and simplify.
Find [latex]{a}_{1}[/latex] by substituting [latex]k=1[/latex] into the given explicit formula.
[latex]{a}_{1}=3\cdot {2}^{1}=6[/latex]
We can see from the given explicit formula that [latex]r=2[/latex]. The upper limit of summation is [latex]6[/latex], so [latex]n=6[/latex].Substitute values for [latex]{a}_{1},r[/latex], and [latex]n[/latex] into the formula, and simplify.