Identifying G.P. Terms

The given series is a Geometric Progression (G.P.): $4/11, 4/121, 4/1331, 4/14641, ...$

• First term, $a = 4/11$.
• Common ratio, $r = \frac{\text{Second term}}{\text{First term}} = \frac{4/121}{4/11} = \frac{4}{121} \times \frac{11}{4} = \frac{11}{121} = \frac{1}{11}$.

Calculating G.P. Sum

The formula for the sum of the first $n$ terms of a G.P. is $S_n = \frac{a(1 - r^n)}{1 - r}$, applicable when $|r| < 1$. In this case, $r = 1/11$, which is less than 1.

Substitute the values of $a$ and $r$ into the formula:

$ S_n = \frac{4/11 \left( 1 - (1/11)^n \right)}{1 - 1/11} $

Simplify the denominator:

$ 1 - \frac{1}{11} = \frac{11 - 1}{11} = \frac{10}{11} $

Now substitute this back into the sum formula:

$ S_n = \frac{4/11 \left( 1 - (1/11)^n \right)}{10/11} $

Simplify the expression:

$ S_n = \frac{4}{11} \times \frac{11}{10} \left( 1 - \left(\frac{1}{11}\right)^n \right) $

$ S_n = \frac{4}{10} \left( 1 - \left(\frac{1}{11}\right)^n \right) $

Reduce the fraction:

$ S_n = \frac{2}{5} \left( 1 - \left(\frac{1}{11}\right)^n \right) $

This matches the expression in Option 1.