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Given-that-f-r-r-1-r-show-that-f-r-f-r-1-r-r-2-1-Hence-or-otherwise-show-that-2-5-3-10-4-17-n-n-2-1-n-1-n-2-

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Question Number 143393 by ZiYangLee last updated on 13/Jun/21
Given that f(r)=(r+1)! ∙ r, show that f(r)−f(r−1)=r!(r^2 +1). Hence or otherwise, show that 2! ∙ 5+3! ∙ 10+4! ∙ 17+......n!(n^2 +1)=(n+1)! ∙ (n−2)
Giventhatf(r)=(r+1)!⋅r,showthatf(r)−f(r−1)=r!(r2+1).Henceorotherwise,showthat2!⋅5+3!⋅10+4!⋅17+……n!(n2+1)=(n+1)!⋅(n−2)
Answered by TheHoneyCat last updated on 13/Jun/21
f(r)−f(r−1) =(r+1)!∙r−r!∙(r−1) =(r+1)∙r!∙r−(r−1)∙r! =r!∙(r^2 +r−r+1) =r!∙(r^2 +1)
f(r)−f(r−1)=(r+1)!⋅r−r!⋅(r−1)=(r+1)⋅r!⋅r−(r−1)⋅r!=r!⋅(r2+r−r+1)=r!⋅(r2+1)

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