The sum of the series   1+ 2x2 + 3x2^2 + 4x2^3 + 5x2^4 +.........+ 100x2^99 is
 
a) 99x 2^100
 
b) 99 x 2^100 +1
 
c) 100 x 2^100
 
d) none of these

Asked by ahuja8087 | 16th Nov, 2015, 04:38: PM

Expert Answer:

begin mathsize 11px style straight S equals 1 space plus space 2 cross times 2 space plus space 3 cross times 2 squared space plus space 4 cross times 2 cubed space plus space... space plus space 100 cross times 2 to the power of 99... left parenthesis 1 right parenthesis 2 straight S equals space space space space space space space space space space space space 2 space plus space 2 cross times 2 squared space plus space 3 cross times 2 cubed space plus space 4 cross times 2 to the power of 4 plus space... space plus 99 cross times 2 to the power of 99 space plus space 100 cross times 2 to the power of 100... left parenthesis 2 right parenthesis Subtracting space left parenthesis 2 right parenthesis space from space left parenthesis 1 right parenthesis comma space we space get minus straight S space equals 1 space plus space 2 space plus space 2 squared space plus space 2 cubed space plus.... space plus space 2 to the power of 99 space minus 100 cross times 2 to the power of 100 rightwards double arrow negative straight S space equals space fraction numerator 2 to the power of 100 minus 1 over denominator 2 minus 1 end fraction minus 100 cross times 2 to the power of 100 rightwards double arrow straight S equals 100 cross times 2 to the power of 100 minus open parentheses 2 to the power of 100 minus 1 close parentheses rightwards double arrow straight S equals 99 cross times 2 to the power of 100 space plus space 1  end style

Answered by satyajit samal | 17th Nov, 2015, 01:15: PM