Finding the Sum of the Geometric Progression

The given series is a Geometric Progression (G.P.): $5/11, 5/121, 5/1331, 5/14641, \dots$

Identify G.P. Parameters

• The first term, $a$, is $5/11$.
• To find the common ratio, $r$, divide the second term by the first term: $r = \frac{5/121}{5/11} = \frac{5}{121} \times \frac{11}{5} = \frac{11}{121} = \frac{1}{11}$
• The number of terms is $n$.

Apply the Sum Formula

The formula for the sum of the first $n$ terms of a G.P. where $|r| < 1$ is:

$ S_n = \frac{a(1-r^n)}{1-r} $

Calculate the Sum

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

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

First, calculate the denominator:

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

Now substitute this back into the sum formula:

$ S_n = \frac{\frac{5}{11} \left(1 - \left(\frac{1}{11}\right)^n\right)}{\frac{10}{11}} $

Simplify the expression:

$ S_n = \frac{5}{11} \times \frac{11}{10} \times \left(1 - \left(\frac{1}{11}\right)^n\right) $ $ S_n = \frac{5}{10} \times \left(1 - \left(\frac{1}{11}\right)^n\right) $ $ S_n = \frac{1}{2} \left(1 - \left(\frac{1}{11}\right)^n\right) $

Conclusion

The sum of the G.P. to $n$ terms is $\frac{1}{2}(1-(\frac{1}{11})^n)$.