Permutations and Combinations

Permutations and Combinations

Question

Sum the following series:

0.5 + 0.55 + 0.555.......... to n terms



Answer

Let space space straight S subscript straight n = 0.5 + 0.55 + 0.555 +............ to n terms
           = 5[0.1 + 0.11 + 0.111 + ........... to n terms]
           = 5 over 9 left square bracket 0.9 space plus space 0.99 space plus space 0.999 space plus space.......... space to space straight n space terms right square bracket
           = 5 over 9 left square bracket left parenthesis 1 minus 0.1 right parenthesis space plus space left parenthesis 1 minus.01 right parenthesis space plus space left parenthesis 1 minus.001 right parenthesis space plus space....... space to space straight n space terms right square bracket
           = 5 over 9 open square brackets straight n minus open parentheses 1 over 10 plus 1 over 100 plus 1 over 1000 plus....... space to space straight n space terms close parentheses close square brackets
           = 5 over 9 open square brackets straight n minus fraction numerator begin display style 1 over 10 end style open parentheses 1 minus begin display style 1 over 10 to the power of straight n end style close parentheses over denominator 1 minus begin display style 1 over 10 end style end fraction close square brackets space equals space 5 over 9 open square brackets straight n minus 1 over 9 open parentheses 1 minus 1 over 10 to the power of straight n close parentheses close square brackets space equals space 5 over 81 open square brackets 9 straight n minus 1 space plus space 1 over 10 to the power of straight n close square brackets
          

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