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Question

An A.P. has even number of terms, the first term of which is unity. S1 and S2 stand for the sum of its first n terms and sum of its last n terms such that S1/S2=λ . What is the common difference of A.P.?

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Solution

Let the common difference is d
And let number of terms is 2n
S1=Sn=n2[2+(n1)d]
and S2=S2n=2n2[2+(2n1)d]
Given S1S2=SnS2nSn=λ ( constant )
S2nSnSn=1λ=k ( say )
S2nSn1=kS2nSn=k+1
2n2[2+(2n1)d]n2[2+(n1)d]=k+1 2[2+(2n1)d][2+(n1)d]=k+1
d=2(k1)n(3k)+k1
Since ratio is independent of n so coefficient of n should be zero
3k=0k=3
So, d=2

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