For -1 < < 1, let (n) denote the sum of the geometric progression 43 + 43r+ 43P + 432
Let'a' between-1 and 1 satisfy s(a). S(-a) = 2021. Find the value of (a) + St-a).
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Sum of the First n Terms of a Geometric Sequence
If a sequence is geometric there are ways to find the sum of the first n terms, denoted Sn, without actually adding all of the terms.
To find the sum of the first Sn terms of a geometric sequence use the formula
Sn=a1(1−rn)1−r,r≠1,
where n is the number of terms, a1 is the first term and r is the common ratio.
The sum of the first n terms of a geometric sequence is called geometric series.
Example 1:
Find the sum of the first 8 terms of the geometric series if a1=1 and r=2.
S8=1(1−28)1−2=255
Example 2:
Find S10 of the geometric sequence 24,12,6,⋯.
First, find r.
r=r2r1=1224=12
Now, find the sum:
S10=24(1−(12)10)1−12=306964
Example 3:
Evaluate.
∑n=1103(−2)n−1
(You are finding S10 for the series 3−6+12−24+⋯, whose common ratio is −2.)
Sn=a1(1−rn)1−rS10=3[1−(−2)10]1−(−2)=3(1−1024)3=−1023