## Which equation could be used to calculate the sum of the geometric series? 1/4+2/9+4/27+8/81+16/243?

Question

Which equation could be used to calculate the sum of the geometric series? 1/4+2/9+4/27+8/81+16/243?

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1. Answer: Sum of the geometric series will be [tex]frac{763}{972}[/tex]

Step-by-step explanation:

Since we have given that

[tex]frac{1}{4}+frac{2}{9}+frac{4}{27}+frac{8}{81}+frac{16}{243}[/tex]

Here,

[tex]a=frac{2}{9}r=frac{a_2}{a_1}r=frac{frac{4}{27}}{frac{2}{9}}=frac{4}{27}times frac{9}{2}=frac{2}{3}n=4[/tex]

As we know the formula for “Sum of n terms in geometric series “:

[tex]S_n=frac{a(1-r^n)}{1-r}S_n=frac{frac{2}{9}(1-frac{2}{3}^4)}{1-frac{2}{3}}S_n=frac{130}{243}[/tex]

So, Complete sum will be

[tex]frac{130}{243}+frac{1}{4}=frac{520+243}{972}=frac{763}{972}[/tex]

Hence, Sum of the geometric series will be [tex]frac{763}{972}[/tex]

2. From the given sequence:
1/4, 2/9, 4,27, 8/81, 16/243
the common ratio is: 2/3
thus the sum of the series will be given by the formula:
Sn=[a(1-r^n)]/(1-r)
plugging the values we obtain:

Sn=[1/4(1-(2/3)^n)]/(1-2/3)
thus the equation that will be used to find the sum is:
Sn=3[1/4-1/4(2/3)^n]
=3/4[1-(2/3)^n]