Find the sum of the series 1 + \(\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5

1.	Find the sum of the series 1 + \(\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+.....\infty\)
Answer» Let S_∞ = 1 + \(\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+.....\infty\) .....(i) \(\frac{1}{5}\)S_∞ = \(\frac{1}{5}+\frac{4}{5^2}+\frac{7}{5^3}+.....\infty\) .....(ii) Subtracting eqn (ii) from eqn (i), we get \(\big(1-\frac{1}{5}\big)S_\infty\) = 1 + \(\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}+.....\infty\) \(\frac{4}{5}\)S_∞ = 1 + 3\(\big(\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+.....\infty\big)\) = 1 + 3 \(\bigg(\frac{\frac{1}{5}}{1-\frac{1}{5}}\bigg)\) (∵ Sum of infinite G.P. = \(\frac{a}{1-r}\)) = 1 + \(\frac{\frac{3}{5}}{\frac{4}{5}}\) = 1 + \(\frac{3}{4}\) = \(\frac{7}{4}\) S_∞= \(\frac{7}{4}\) x \(\frac{5}{4}\) = \(\frac{35}{16}.\)

Find the sum of the series 1 + \(\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+.....\infty\)

Answer»

Let S_∞ = 1 + \(\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+.....\infty\) .....(i)

\(\frac{1}{5}\)S_∞ = \(\frac{1}{5}+\frac{4}{5^2}+\frac{7}{5^3}+.....\infty\) .....(ii)

Subtracting eqn (ii) from eqn (i), we get

\(\big(1-\frac{1}{5}\big)S_\infty\) = 1 + \(\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}+.....\infty\)

\(\frac{4}{5}\)S_∞ = 1 + 3\(\big(\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+.....\infty\big)\)

= 1 + 3 \(\bigg(\frac{\frac{1}{5}}{1-\frac{1}{5}}\bigg)\) (∵ Sum of infinite G.P. = \(\frac{a}{1-r}\))

= 1 + \(\frac{\frac{3}{5}}{\frac{4}{5}}\) = 1 + \(\frac{3}{4}\) = \(\frac{7}{4}\)

S_∞= \(\frac{7}{4}\) x \(\frac{5}{4}\) = \(\frac{35}{16}.\)