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Question

1) For an A.P, show that tm+t2n+m=2tm+n

2) If the sum of p terms of an A.P is q and the sum of its q terms is p, then find the sum Of its (p+q) term.

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Solution

1)
tm=a+(m1)d
t2n+m=a+(2n+m1)d
2tm+n=2[a+(n+m1)d]
Adding both,
tm+t2n+m =a+(m1)d+a+(2n+m1),d=2a+(2n+m1+m1)d,=2a+(2n+2m2)d,=2[a+(n+m1)d]. (Proved)
2)
Sp=q, Sq=p
q=p2{2a+(p1)d}p=q2{2a+(q1)d}
Subtracting both.
qp=p2{2a+(p1)d}q2{2a+(q1)d}
qp=a(pq)+d2(p2pq2+q)
qp=a(pq)+d2(pq)(p+q1)
Dividing (pq) from both sides.
1=a+d2(p+q1)
2=2a+(p+q1)d
Sp+q=p+q2{2a+(p+q1)d}
Sp+q=p+q2{2}=(p+q)

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