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Question

Given the ratio of sum of n terms of two AP's = (7n+1):(4n+27). Find the ratio of their mth terms.

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Solution

Given ratio of sum of n terms of two AP's = (7n+1):(4n+27)
Let's consider the ratio these two AP's mth terms as am:am (2)
Recall the nth term of AP formula, an=a+(n1)d
Hence, equation (2) becomes,
am:am=a+(m1)d:a+(m1)d
On multiplying by 2, we get
am:am=[2a+2(md)d]:[2a+2(m1)d]
=[2a+{(2m1)1}d]:[2a+{(2m1)1}d]
=S2m1:S2m1
=[7(2m1)+1]:[4(2m1)+27] [from (1)]
=[14m7+1]:[8m4+27]
=[14m6]:[8m+23]
Thus the ratio of mth terms of two AP's is [14m - 6] : [8m + 23].

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