Find the sum of 50 terms of the series 9 + 9.9 + 99.9 + 999.9 + …

Asked by Jagan Joseph | 4th Nov, 2014, 11:59: PM

Expert Answer:

C o n s i d e r space t h e space g i v e n space s e r i e s : S equals 9 plus 9.9 plus 99.9 plus 999.9 plus.... u p space t o space 50 space t e r m s R e w r i t i n g space t h e space a b o v e space s e r i e s comma space w e space h a v e comma S equals 9 over 10 to the power of 0 plus 99 over 10 to the power of 1 plus 999 over 10 squared plus 9999 over 10 cubed plus... plus fraction numerator 99...9 over denominator 10 to the power of 49 end fraction rightwards double arrow S equals fraction numerator 10 to the power of 1 minus 1 over denominator 10 to the power of 0 end fraction plus fraction numerator 10 squared minus 1 over denominator 10 to the power of 1 end fraction plus fraction numerator 10 cubed minus 1 over denominator 10 squared end fraction plus fraction numerator 10 to the power of 4 minus 1 over denominator 10 cubed end fraction plus... plus fraction numerator 10 to the power of 50 minus 1 over denominator 10 to the power of 49 end fraction rightwards double arrow S equals 10 to the power of 1 over 10 to the power of 0 minus 1 over 10 to the power of 0 plus 10 squared over 10 to the power of 1 minus 1 over 10 to the power of 1 plus 10 cubed over 10 squared minus 1 over 10 squared plus 10 to the power of 4 over 10 cubed minus 1 over 10 cubed plus... plus 10 to the power of 50 over 10 to the power of 49 minus 1 over 10 to the power of 49 rightwards double arrow S equals 10 to the power of 1 minus 0 end exponent minus 1 over 10 to the power of 0 plus 10 to the power of 2 minus 1 end exponent minus 1 over 10 to the power of 1 plus 10 to the power of 3 minus 2 end exponent minus 1 over 10 squared plus 10 to the power of 4 minus 3 end exponent minus 1 over 10 cubed plus... plus 10 to the power of 50 minus 49 end exponent minus 1 over 10 to the power of 49 rightwards double arrow S equals 10 minus 1 over 10 to the power of 0 plus 10 minus 1 over 10 to the power of 1 plus 10 minus 1 over 10 squared plus 10 minus 1 over 10 cubed plus... plus 10 minus 1 over 10 to the power of 49 rightwards double arrow S equals 10 plus 10 plus 10 plus... plus 10 minus 1 over 10 to the power of 0 minus 1 over 10 to the power of 1 minus 1 over 10 squared minus 1 over 10 cubed minus... minus 1 over 10 to the power of 49 rightwards double arrow S equals 50 cross times 10 minus open square brackets 1 over 10 to the power of 0 plus 1 over 10 to the power of 1 plus 1 over 10 squared plus 1 over 10 cubed plus... plus 1 over 10 to the power of 49 close square brackets rightwards double arrow S equals 500 minus open square brackets 1 plus 1 over 10 to the power of 1 plus 1 over 10 squared plus 1 over 10 cubed plus... plus 1 over 10 to the power of 49 close square brackets N o w space c o n s i d e r comma space t h e space s u m space 1 plus 1 over 10 to the power of 1 plus 1 over 10 squared plus 1 over 10 cubed plus... plus 1 over 10 to the power of 49 I t space i s space a space G P. T h u s comma S subscript 1 equals space 1 plus 1 over 10 to the power of 1 plus 1 over 10 squared plus 1 over 10 cubed plus... plus 1 over 10 to the power of 49 rightwards double arrow S subscript 1 equals fraction numerator 1 minus open parentheses begin display style 1 over 10 end style close parentheses to the power of 50 over denominator 1 minus begin display style 1 over 10 end style end fraction equals fraction numerator 10 to the power of 50 minus 1 over denominator 9 cross times 10 to the power of 49 end fraction T h u s comma S equals 500 minus fraction numerator 10 to the power of 50 minus 1 over denominator 9 cross times 10 to the power of 49 end fraction space space space space space space space space space space space space space space

Answered by Vimala Ramamurthy | 5th Nov, 2014, 09:10: AM