The sum of the infinite series
$$1 + {2 \over 3} + {7 \over {{3^2}}} + {{12} \over {{3^3}}} + {{17} \over {{3^4}}} + {{22} \over {{3^5}}} + ......$$ is equal to :
Solution
$$S = 1 + {2 \over 3} + {7 \over {{3^2}}} + {{12} \over {{3^3}}} + {{17} \over {{3^4}}} + ....$$<br><br>$${S \over 3} = {1 \over 3} + {2 \over {{3^2}}} + {7 \over {{3^3}}} + {{12} \over {{3^4}}} + ....$$<br><br>$$2S = 1 + {1 \over 3} + {5 \over {{3^2}}} + {5 \over {{3^3}}} + {5 \over {{3^4}}} + ....$$ + up to infinite terms<br><br>$${{2S} \over 3} = {4 \over 3} + {5 \over 3}\left\{ {{{1/3} \over {1 - {1 \over 3}}}} \right\} = {5 \over 6} + {4 \over 3} = {{13} \over 6}$$
<br><br>$\Rightarrow$ S = ${13 \over 4}$
About this question
Subject: Mathematics · Chapter: Sequences and Series · Topic: Arithmetic Progression
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