Answer:
S 5 = StartFraction one-third (1 minus (two-thirds) Superscript 5 Baseline) Over (1 minus two-thirds) EndFraction
Explanation:
Given the geometric series:
1/3+2/9+4/27+8/81+16/243
First we must know that the series is a finite series with just 5terms.
Before we can know the formula to calculate sum of the first five terms of the series, we must determine its common ratio (r) first.
r = (2/9)÷1/3 = 4/27÷2/9= 8/81÷4/27
r = 2/9 × 3/1
r = 2/3
Similarly;
r = 4/27×9/2
r = 2/3
Since all values of r is the as them the common ratio is 2/3.
If r< 1 in geometric series, then the formula for finding its sum is applicable
Sn = a(1-rⁿ)/1-r
a is the first term = 1/3
r is the common ratio = 2/3
n is the number of terms = 5
Substituting the values in the formula we have:
S5 = 1/3{1-(2/3)^5}/1-2/3
This gives the requires equation
S 5 = StartFraction one-third (1 minus (two-thirds) Superscript 5 Baseline) Over (1 minus two-thirds) EndFraction