If $$S = {7 \over 5} + {9 \over {{5^2}}} + {{13} \over {{5^3}}} + {{19} \over {{5^4}}} + ....$$, then 160 S is equal to ________.
Answer (integer)
305
Solution
$$S = {7 \over 5} + {9 \over {{5^2}}} + {{13} \over {{5^3}}} + {{19} \over {{5^4}}} + ....$$<br><br>${1 \over 5}S = {7 \over 5} + {9 \over {{5^3}}} + {{13} \over {{5^4}}} + ....$<br><br>On subtracting<br><br>$${4 \over 5}S = {7 \over 5} + {2 \over {{5^2}}} + {4 \over {{5^3}}} + {6 \over {{5^4}}} + ....$$<br><br>$$S = {7 \over {14}} + {1 \over {10}}\left( {1 + {2 \over 5} + {3 \over {{5^2}}} + ...} \right)$$<br><br>$S = {7 \over 4} + {1 \over {10}}{\left( {1 - {1 \over 5}} \right)^{ - 2}}$<br><br>$= {7 \over 4} + {1 \over {10}} \times {{25} \over {16}} = {{61} \over {32}}$<br><br>$\Rightarrow$ 160S = 5 $\times$ 61 = 305
About this question
Subject: Mathematics · Chapter: Sequences and Series · Topic: Arithmetic Progression
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