Question
The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, . . . . . , is
A.
$$\frac{7}{{81}}\left( {179 - {{10}^{ - 20}}} \right)$$
B.
$$\frac{7}{{9}}\left( {99 - {{10}^{ - 20}}} \right)$$
C.
$$\frac{7}{{81}}\left( {179 + {{10}^{ - 20}}} \right)$$
D.
$$\frac{7}{{9}}\left( {99 + {{10}^{ - 20}}} \right)$$
Answer :
$$\frac{7}{{81}}\left( {179 + {{10}^{ - 20}}} \right)$$
Solution :
Given sequence can be written as
$$\eqalign{
& \frac{7}{{10}} + \frac{{77}}{{100}} + \frac{{777}}{{{{10}^3}}} + ..... + {\text{ up to 20 terms}} \cr
& = {\text{7}}\left[ {\frac{1}{{10}} + \frac{{11}}{{100}} + \frac{{111}}{{{{10}^3}}} + ..... + {\text{up to 20 terms}}} \right] \cr
& {\text{Multiply and divide by 9}} \cr
& = \frac{7}{9}\left[ {\frac{9}{{10}} + \frac{{99}}{{100}} + \frac{{999}}{{{{10}^3}}} + ..... + {\text{up to 20 terms}}} \right] \cr
& = \frac{7}{9}\left[ {\left( {1 - \frac{1}{{10}}} \right) + \left( {1 - \frac{1}{{{{10}^2}}}} \right) + \left( {1 - \frac{1}{{{{10}^3}}}} \right) + ..... + {\text{up to 20 terms}}} \right] \cr
& = \frac{7}{9}\left[ {20 - \frac{{\frac{1}{{10}}\left( {1 - {{\left( {\frac{1}{{10}}} \right)}^{20}}} \right)}}{{1 - \frac{1}{{10}}}}} \right] \cr
& = \frac{7}{9}\left[ {\frac{{179}}{9} + \frac{1}{9}{{\left( {\frac{1}{{10}}} \right)}^{20}}} \right] \cr
& = \frac{7}{{81}}\left[ {179 + {{\left( {10} \right)}^{ - 20}}} \right] \cr} $$