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please explain each step, thanks very much

 Jun 5, 2019
 #1
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sequence proof

 

Let y=1+x or x=y1 

f(x)=1y+2y2+3y3+4y4++n1yn1+nyn|yyf(x)=1+2y+3y2+4y3+5y4++nyn1yf(x)f(x)=1+1y+1y2+1y3+1y4++1yn1nyn(y1)f(x)=1+1y+1y2+1y3+1y4++1yn1nyn|y1=xxf(x)=1+1y+1y2+1y3+1y4++1yn1nyn

 

xf(x)=1+1y+1y2+1y3+1y4++1yn1=s (sum of a geometric progression)nynxf(x)=snyns=1+1y+1y2+1y3+1y4++1yn1|:ysy=1y+1y2+1y3+1y4++1yn1+1ynssy=11yns(11y)=11yns=11yn11yxf(x)=11yn11ynyn|y=1+xxf(x)=11(1+x)n111+xn(1+x)n

 

xf(x)=11(1+x)n111+xn(1+x)nxf(x)=(1+x)n1(1+x)n1+x11+xn(1+x)nxf(x)=(1+x)n1(1+x)nx1+xn(1+x)nxf(x)=((1+x)n1)(1+x)x(1+x)nn(1+x)nxxxf(x)=((1+x)n1)(1+x)nxx(1+x)n|:xf(x)=((1+x)n1)(1+x)nxx2(1+x)nf(x)=(1+x)n+1(1+x)nxx2(1+x)n

 

laugh

 Jun 5, 2019

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