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

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]

Answer Link

RenatoMattice RenatoMattice · Answer 2 · 2018-12-17T06:20:39+01:00

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]