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If p,q,r are in arithmetic progression then show that p2(q+r)+ q2(r+p)+ r2(p+q )=2/9(p+q+r)3
Asked by rushabhjain.a | 20 Nov, 2018, 11:13: AM
we need to prove,  p2(q+r) + q2(r+p) + r2(p+q) = (2/9) ( p+q+r )3

let us substitute in LHS q = p+d and r=p+2d , where d is the common difference of A.P.

Then LHS = p2(2p+3d) + 2(p+d)2(p+d) + (p+2d)2(2p+d) = 6p3 + 18pd2 + 18pd2 + 6d3

RHS = (2/9)×(p+p+d+p+2d)3 = 6(p+d)3 = 6p3 + 18pd2 + 18pd2 + 6d3

Hence we proved LHS = RHS
Answered by Thiyagarajan K | 22 Nov, 2018, 11:18: PM

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