Question

Find the sum to infinity of the G.P.:
 $ 5,\dfrac{{20}}{7},\dfrac{{80}}{{49}},..... $

Answer 1

Hint: For this type of problem in which it is given that series is infinite G.P. there from the given G.P. we first find the first term and common ratio of given G.P. series and then using these values in a formula of sum of infinite G.P. to find its value or required value or solution of a given problem.
Formulas used: Sum of infinite G.P. $ {S_\infty } = \dfrac{a}{{1 - r}} $ , where ‘a’ and ‘r’ are first term and common ratio of given geometric progression.

Complete step-by-step answer:
Given series is $ 5,\dfrac{{20}}{7},\dfrac{{80}}{{49}},..... $
It is also given that the above series is a geometric progression.
Therefore, first term of series is $ 5 $
and common ratio (r) will be given as: $ \dfrac{{\dfrac{{20}}{7}}}{5} = \dfrac{4}{7} $
Since, it is given that there are infinite terms in a given series.
So, the series is infinite G.P.
Therefore, we use infinite G.P. formula to find its sum.
Sum of infinite G.P. is given as $ {S_\infty } = \dfrac{a}{{1 - r}} $
Substituting values in above formula. We have,
 $ {S_\infty } = \dfrac{5}{{1 - \dfrac{4}{7}}} $
Simplifying the right hand side of the above formed equation. We have,
 $
  {S_\infty } = \dfrac{5}{{\dfrac{{7 - 4}}{7}}} \\
   \Rightarrow {S_\infty } = \dfrac{5}{{\dfrac{3}{7}}} \\
   \Rightarrow {S_\infty } = 5 \times \dfrac{7}{3} \\
   \Rightarrow {S_\infty } = \dfrac{{35}}{3} \;
  $
Hence, from above we see that sum of infinite geometric progression $ 5,\dfrac{{20}}{7},\dfrac{{80}}{{49}},..... $ is $ \dfrac{{35}}{3} $ .
So, the correct answer is “$ \dfrac{{35}}{3} $”.

Note: We know that in G.P. series there are different sum formulas like $ {S_n} = a\left( {\dfrac{{1 - {r^n}}}{{1 - r}}} \right),r < 1\,,\,\,{S_n} = a\left( {\dfrac{{{r^n} - 1}}{{r - }}} \right),r > 1 $ and $ {S_\infty } = \dfrac{a}{{1 - r}} $ . So one must choose the correct formula related to a given problem or we can say according to the conditions given in the problem to find the correct value of the given problem and hence the required solution of the problem.