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If-the-sum-of-first-p-terms-first-q-terms-and-first-r-terms-of-an-AP-be-x-y-and-z-respectively-Then-x-p-q-r-y-q-r-p-z-r-p-q-is-

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Question Number 11979 by 786786AM last updated on 08/Apr/17
If the sum of first p terms, first q terms and first r terms of an AP be x, y and z respectively. Then (x/p)(q−r) + (y/q)(r−p) + (z/r)(p−q) is
Ifthesumoffirstpterms,firstqtermsandfirstrtermsofanAPbex,yandzrespectively.Thenxp(q−r)+yq(r−p)+zr(p−q)is
Answered by ajfour last updated on 09/Apr/17
(x/p)=a+(p−1)(d/2) (y/q)=a+(q−1)(d/2) (z/r)=a+(r−1)(d/2) so, (x/p)(q−r)+(y/q)(r−p)+(z/r)(p−q) =( a−(d/2))(q−r+r−p+p−q) +(d/2)[p(q−r)+q(r−p)+r(p−q)] =(a−(d/2))(0) + (d/2)(0) =0 .
xp=a+(p−1)d2yq=a+(q−1)d2zr=a+(r−1)d2so,xp(q−r)+yq(r−p)+zr(p−q)=(a−d2)(q−r+r−p+p−q)+d2[p(q−r)+q(r−p)+r(p−q)]=(a−d2)(0)+d2(0)=0.

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