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Question

If the ratio of the sum of first n terms of two A.P's is (7n+1): (4n+27), find the of their mth terms

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Solution

Given ratio of sum of n terms of two AP's =(7n+1):(4n+27)
n2(2a+(n1)d):n2(2a+(n1)d)=(7n+1):(4n+27)(2a+(n1)d):(2a+(n1)d)=(7n+1):(4n+27)(1)
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(m1)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]:[8m+23]=[14m6]:[8m+23]
Thus the ratio of mth terms of two AP's is [14m-6]:[8m+23].

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