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Permutations And Combinations

Question

Sum the series

$space space 2 over 9 space plus space 1 third plus space 1 half plus space............ space plus space 81 over 32$

Solution

The given series is : $WiredFaculty$
Here, $space space straight a equals 2 over 9 comma space space straight r space equals space fraction numerator begin display style 1 third end style over denominator begin display style 2 over 9 end style end fraction space equals space 1 third.9 over 2 equals 3 over 2.$
Let $l space equals space 81 over 32$ be the nth term
Required sum = $fraction numerator straight a minus lr over denominator 1 minus straight r end fraction space equals space fraction numerator begin display style 2 over 9 end style minus begin display style 81 over 32 end style open parentheses begin display style 3 over 2 end style close parentheses over denominator 1 minus begin display style 3 over 2 end style end fraction space equals space fraction numerator begin display style 2 over 9 end style minus begin display style 243 over 64 end style over denominator begin display style fraction numerator negative 1 over denominator 2 end fraction end style end fraction space equals space fraction numerator 128 minus 2187 over denominator 9.64 end fraction open parentheses fraction numerator negative 2 over denominator 1 end fraction close parentheses space equals space fraction numerator 2059 over denominator 9. space 32 space end fraction space equals space 2059 over 288$

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Permutations And Combinations

Sum the series

$space space 2 over 9 space plus space 1 third plus space 1 half plus space............ space plus space 81 over 32$

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