If a sequence is geometric there are ways to find the sum of the first <span>nn</span> terms, denoted <span><span>Sn</span><span>Sn</span></span>, without actually adding all of the terms.
To find the sum of the first <span><span>Sn</span><span>Sn</span></span> terms of a geometric sequence use the formula
<span><span><span>Sn</span>=<span><span><span>a1</span>(1−<span>rn</span>)</span><span>1−r</span></span>,r≠1</span><span><span>Sn</span>=<span><span><span>a1</span>(1−<span>rn</span>)</span><span>1−r</span></span>,r≠1</span></span>,
where <span>nn</span> is the number of terms, <span><span>a1</span><span>a1</span></span> is the first term and <span>rr</span> is the common ratio.
<span>The sum of the first </span><span>nn</span><span> terms of a geometric sequence is called geometric series.</span>
Example 1:
Find the sum of the first 8 terms of the geometric series if <span><span><span>a1</span>=1</span><span><span>a1</span>=1</span></span> and <span><span>r=2</span><span>r=2</span></span>.
<span><span><span>S8</span>=<span><span>1(1−<span>28</span>)</span><span>1−2</span></span>=255</span><span><span>S8</span>=<span><span>1(1−<span>28</span>)</span><span>1−2</span></span>=255</span></span>
Example 2:
Find <span><span>S10</span><span>S10</span></span> of the geometric sequence <span><span>24,12,6,⋯</span><span>24,12,6,⋯</span></span>.
First, find <span>rr</span>.
<span><span>r=<span><span>r2</span><span>r1</span></span>=<span>1224</span>=<span>12</span></span><span>r=<span><span>r2</span><span>r1</span></span>=<span>1224</span>=<span>12</span></span></span>
Now, find the sum:
<span><span><span>S10</span>=<span><span>24<span>(<span>1−<span><span>(<span>12</span>)</span>10</span></span>)</span></span><span>1−<span>12</span></span></span>=<span>306964</span></span><span><span>S10</span>=<span><span>24<span>(<span>1−<span><span>(<span>12</span>)</span>10</span></span>)</span></span><span>1−<span>12</span></span></span>=<span>306964</span></span></span>
Example 3:
Evaluate.
<span><span><span>∑<span>n=1</span>10</span><span>3<span><span>(−2)</span><span>n−1</span></span></span></span><span><span>∑<span>n=1</span>10</span><span>3<span><span>(−2)</span><span>n−1</span></span></span></span></span>
(You are finding <span><span>S10</span><span>S10</span></span> for the series <span><span>3−6+12−24+⋯</span><span>3−6+12−24+⋯</span></span>, whose common ratio is <span><span>−2</span><span>−2</span></span>.)
<span><span><span><span>Sn</span>=<span><span><span>a1</span>(1−<span>rn</span>)</span><span>1−r</span></span></span><span><span>S10</span>=<span><span>3<span>[<span>1−<span><span>(−2)</span>10</span></span>]</span></span><span>1−(−2)</span></span>=<span><span>3(1−1024)</span>3</span>=−1023</span></span><span><span><span>Sn</span>=<span><span><span>a1</span>(1−<span>rn</span>)</span><span>1−r</span></span></span><span><span>S10</span>=<span><span>3<span>[<span>1−<span><span>(−2)</span>10</span></span>]</span></span><span>1−(−2)</span></span>=<span><span>3(1−1024)</span>3</span>=−1023</span></span></span>
In order for an infinite geometric series to have a sum, the common ratio <span>rr</span> must be between <span><span>−1</span><span>−1</span></span> and <span>11</span>. Then as <span>nn</span> increases, <span><span>rn</span><span>rn</span></span> gets closer and closer to <span>00</span>. To find the sum of an infinite geometric series having ratios with an absolute value less than one, use the formula, <span><span>S=<span><span>a1</span><span>1−r</span></span></span><span>S=<span><span>a1</span><span>1−r</span></span></span></span>, where <span><span>a1</span><span>a1</span></span> is the first term and <span>rr</span> is the common ratio.
Example 4:
Find the sum of the infinite geometric sequence
<span><span>27,18,12,8,⋯</span><span>27,18,12,8,⋯</span></span>.
First find <span>rr</span>:
<span><span>r=<span><span>a2</span><span>a1</span></span>=<span>1827</span>=<span>23</span></span><span>r=<span><span>a2</span><span>a1</span></span>=<span>1827</span>=<span>23</span></span></span>
Then find the sum:
<span><span>S=<span><span>a1</span><span>1−r</span></span></span><span>S=<span><span>a1</span><span>1−r</span></span></span></span>
<span><span>S=<span>27<span>1−<span>23</span></span></span>=81</span><span>S=<span>27<span>1−<span>23</span></span></span>=81</span></span>
Example 5:
Find the sum of the infinite geometric sequence
<span><span>8,12,18,27,⋯</span><span>8,12,18,27,⋯</span></span> if it exists.
First find <span>rr</span>:
<span><span>r=<span><span>a2</span><span>a1</span></span>=<span>128</span>=<span>32</span></span><span>r=<span><span>a2</span><span>a1</span></span>=<span>128</span>=<span>32</span></span></span>
Since <span><span>r=<span>32</span></span><span>r=<span>32</span></span></span> is not less than one the series has no sum.