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Let the terms a(1),a(2),a(3),…a(n) be in...

Let the terms `a_(1),a_(2),a_(3),…a_(n)` be in G.P. with common ratio r. Let `S_(k)` denote the sum of first k terms of this G.P.. Prove that `S_(m-1)xxS_(m)=(r+1)/r`SigmaSigma_(i le itj le n)a_(i)a_(j)`

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We have `a_(1)+a_(2)+…+a_(m))^(2)`
`=a_(1)^(2)+a_(2)^(2)+…+a_(m)^(2)+2(a_(1)a_(2)+a_(2)a_(3)+…)`
`therefore[(a_(1)(1-r^(m)))/(1-r)]^(2)=(a_(1)^(2)(1-r^(2m)))/(1-r^(2))+2underset(1leiltjlen)(SigmaSigma)a_(i)a_(j)`
`rArr2underset(1leiltjlen)(SigmaSigma)a_(i)a_(j)=(a_(1)^(2)(1-r^(m))^(2))/((1-r)^(2))-(a_(1)^(2)(1-r^(2m)))/(1-r^(2))`
`=(a_(1)^(2)(1-r^(m)))/((1-r)^(2)(1+r))2(r-r^(m))`
`=(2r)/(1+r){a_(1)cdot((1-r^(m-1)))/(1-r)}{(a_(1)(1-r^(m)))/(1-r)}`
`=S_(m-1)xxS_(m)`
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