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Determine-S-S-a-2-ar-a-r-ar-2-a-2r-ar-n-1-a-n-1-r-

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Question Number 5742 by Rasheed Soomro last updated on 26/May/16
•Determine S S=a^2 +ar(a+r)+ar^2 (a+2r)+...+ar^(n−1) {a+(n−1) r}.
∙DetermineSS=a2+ar(a+r)+ar2(a+2r)+…+arn−1{a+(n−1)r}.
Answered by Yozzii last updated on 26/May/16
S=a^2 +ar(a+r)+ar^2 (a+2r)+ar^3 (a+3r)...+ar^(n−1) {a+(n−1)r} rS=a^2 r+ar^2 (a+r)+ar^3 (a+2r)+ar^4 (a+4r)+...+ar^(n−1) (a+(n−2)r)+ar^n (a+(n−1)r) rS=0+(a^2 r)+0+(a^2 r^2 )+0+(ar^3 )+0+(a^2 r^3 )+(2ar^4 )+a^2 r^4 +4ar^5 +...+(a^2 r^(n−1) )+(n−2)ar^n +a^2 r^n +ar^(n+1) (n−1) S=a^2 +(a^2 r)+ar^2 +(a^2 r^2 )+(2ar^3 )+(a^2 r^3 )+(3ar^4 )+...+(a^2 r^(n−1) )+(n−1)ar^n ⇒(r−1)S=−a^2 −ar^2 −ar^3 −ar^4 −...−ar^n −ar^(n+1) +a^2 r^n +nar^(n+1) (r−1)S=a^2 r^n −a^2 −ar^2 (1+r+r^2 +...+r^(n−2) +r^(n−1) )+nar^(n+1) (r−1)S=a^2 (r^n −1)+nar^(n+1) −((ar^2 (r^n −1))/((r−1))) S=((r^n −1)/(r−1))(a^2 −((ar^2 )/(r−1)))+((nar^(n+1) )/(r−1)) S=((nar^(n+1) )/(r−1))+((a(r^n −1)(a(r−1)−r^2 ))/((r−1)^2 )) S=((nar^(n+1) +a^2 (r^n −1))/(r−1))−((ar^2 (r^n −1))/((r−1)^2 )) If r<1 S can be divergent e.g let r=−56.
S=a2+ar(a+r)+ar2(a+2r)+ar3(a+3r)…+arn−1{a+(n−1)r}rS=a2r+ar2(a+r)+ar3(a+2r)+ar4(a+4r)+…+arn−1(a+(n−2)r)+arn(a+(n−1)r)rS=0+(a2r)+0+(a2r2)+0+(ar3)+0+(a2r3)+(2ar4)+a2r4+4ar5+…+(a2rn−1)+(n−2)arn+a2rn+arn+1(n−1)S=a2+(a2r)+ar2+(a2r2)+(2ar3)+(a2r3)+(3ar4)+…+(a2rn−1)+(n−1)arn⇒(r−1)S=−a2−ar2−ar3−ar4−…−arn−arn+1+a2rn+narn+1(r−1)S=a2rn−a2−ar2(1+r+r2+…+rn−2+rn−1)+narn+1(r−1)S=a2(rn−1)+narn+1−ar2(rn−1)(r−1)S=rn−1r−1(a2−ar2r−1)+narn+1r−1S=narn+1r−1+a(rn−1)(a(r−1)−r2)(r−1)2S=narn+1+a2(rn−1)r−1−ar2(rn−1)(r−1)2Ifr<1Scanbedivergente.gletr=−56.
Commented by Rasheed Soomro last updated on 26/May/16
ThankS!
ThankS!

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