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Question Number 141643 by Gbenga last updated on 21/May/21

Answered by qaz last updated on 22/May/21

S=Σ_(n=0) ^∞ (((5n+1)^2 )/(16^n )) =(5xD+1)^2 ∣_(x=1/16) Σ_(n=0) ^∞ x^n =[25(xD)^2 +10xD+1]_(x=1/16) (1/(1−x)) =[25x^2 D^2 +35xD+1]_(x=1/16) (1/(1−x))...................(xD)^2 =x^2 D^2 +xD =[((50x^2 )/((1−x)^3 ))+((35x)/((1−x)^2 ))+(1/(1−x))]_(x=1/16) =((50×16)/(15^3 ))+((35×16)/(15^2 ))+((16)/(15)) =((512)/(135))

S=∑∞n=0(5n+1)216n=(5xD+1)2∣x=1/16∑∞n=0xn=[25(xD)2+10xD+1]x=1/1611−x=[25x2D2+35xD+1]x=1/1611−x...................(xD)2=x2D2+xD=[50x2(1−x)3+35x(1−x)2+11−x]x=1/16=50×16153+35×16152+1615=512135

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