Question

What is the sum of the series \[0.5{\rm{ }} + {\rm{ }}0.55{\rm{ }} + {\rm{ }}0.555{\rm{ }} + ...{\rm{ }}to{\rm{ }}n\] terms?
A) \[\dfrac{5}{9}[n - \dfrac{2}{9}(1 - \dfrac{1}{{{{10}^n}}})]\]
B) \[\dfrac{1}{9}[5 - \dfrac{2}{9}(1 - \dfrac{1}{{{{10}^n}}})]\]
C) \[\dfrac{1}{9}[n - \dfrac{5}{9}(1 - \dfrac{1}{{{{10}^n}}})]\]
D) \[\dfrac{5}{9}[n - \dfrac{1}{9}(1 - \dfrac{1}{{{{10}^n}}})]\]

Answer 1

Hint: in this question we have to find sum of n term of given series. First Rearrange the given series to find given series is in AP or it is in GP Once we get type of series, just apply the formula of sum of series to get required value.

Formula Used: We can find sum of n terms of GP by using
\[{S_n} = \dfrac{{a(1 - {r^n})}}{{1 - r}}\]
Where
\[{S_n}\]is sum of n terms of GP
a is first term of GP
n numbers of terms

Complete step by step solution: \[0.5{\rm{ }} + {\rm{ }}0.55{\rm{ }} + {\rm{ }}0.555{\rm{ }} + ...{\rm{ }}to{\rm{ }}n\]
Now rearrange the above series
\[5[0.1{\rm{ }} + {\rm{ }}0.11{\rm{ }} + {\rm{ }}0.111{\rm{ }} + ...{\rm{ }}to{\rm{ }}n]\]
\[\dfrac{5}{9}[0.9{\rm{ }} + {\rm{ }}0.99{\rm{ }} + {\rm{ }}0.999{\rm{ }} + ...{\rm{ }}to{\rm{ }}n]\]
\[\dfrac{5}{9}[\dfrac{9}{{10}}{\rm{ }} + {\rm{ }}\dfrac{{99}}{{100}}{\rm{ }} + {\rm{ }}\dfrac{{999}}{{1000}}{\rm{ }} + ...{\rm{ }}to{\rm{ }}n]\]
\[\dfrac{5}{9}[(1 - \dfrac{1}{{10}}){\rm{ }} + {\rm{ (1 - }}\dfrac{1}{{100}}{\rm{) }} + {\rm{ (1 - }}\dfrac{1}{{1000}}{\rm{) }} + ...{\rm{ }}to{\rm{ }}n]\]
\[\dfrac{5}{9}[(1 - \dfrac{1}{{10}}){\rm{ }} + {\rm{ (1 - }}\dfrac{1}{{{{10}^2}}}{\rm{) }} + {\rm{ (1 - }}\dfrac{1}{{{{10}^3}}}{\rm{) }} + ...{\rm{ }}to{\rm{ }}n]\]
We can find sum of n terms of GP by using
\[{S_n} = \dfrac{{a(1 - {r^n})}}{{1 - r}}\]
Where
\[{S_n}\]is sum of n terms of GP
a is first term of GP
n numbers of terms
\[\dfrac{5}{9}[n - (\dfrac{1}{{10}}{\rm{ }} + {\rm{ }}\dfrac{1}{{{{10}^2}}}{\rm{ }} + {\rm{ }}\dfrac{1}{{{{10}^3}}}{\rm{ }} + ...{\rm{ }}\dfrac{1}{{{{10}^n}}})]\]
\[\dfrac{5}{9}[n - \dfrac{1}{{10}}{\rm{ (}}\dfrac{{1 - {{(\dfrac{1}{{10}})}^n}}}{{1 - \dfrac{1}{{10}}}})]\]
On simplification we get sum of the given series
\[\dfrac{5}{9}[n - \dfrac{1}{9}{\rm{ (}}1 - \dfrac{1}{{{{10}^n}}})]\]

Option ‘D’ is correct

Note: Whenever given series doesn’t follow any pattern then we first try to rearrange the series. If we get any pattern then follow that pattern to get required values. Sometimes by using pattern we are able to find the first term and common ratio therefore always try to find first term and common ratio if required. Then apply the formula to get the required value.
Sometime students get confused in between AP and GP the only difference in between them is in AP we talk about common difference whereas in GP we talk about common ratio.