The sum of the first 20 terms of the sequence 0.7, 0.77, 0.777, is:

Question

The sum of the first 20 terms of the sequence $0.7, 0.77, 0.777,\ldots$ is:

Accepted Answer

Given sequence can be written as

$${7 \over {10}} + {{77} \over {100}} + {{777} \over {{{10}^3}}} + ..... + $$ up to $20$ terms

$$ = 7\left[ {{1 \over {10}} + {{11} \over {100}} + {{111} \over {{{10}^3}}} + ...... + } \right.\,\,$$ up to $20$ terms ]

Multiply and divide by $9$

$$ = {7 \over 9}\left[ {{9 \over {10}} + {{99} \over {100}} + {{999} \over {1000}} + ......} \right.\,\,$$ $+$ up to $20$ terms ]

$$ = {7 \over 9}\left[ {\left( {1 - {1 \over {10}}} \right)} \right. + \left( {1 - {1 \over {{{10}^2}}}} \right) + \left( {1 - {1 \over {{{10}^3}}}} \right) + ......$$ $+$ up to  $20$ terms ]

$$ = {7 \over 9}\left[ {20 - {{{1 \over {10}}\left( {1 - {{\left( {{1 \over {10}}} \right)}^{20}}} \right)} \over {1 - {1 \over {10}}}}} \right]$$

$$ = {7 \over 9}\left[ {{{179} \over 9} + {1 \over 9}{{\left( {{1 \over {10}}} \right)}^{20}}} \right]$$

$$ = {7 \over {81}}\left[ {179 + {{\left( {10} \right)}^{ - 20}}} \right]$$

Solutions · 1