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Question

The ratio of the sum of n terms of two A.P.’s is (7n+1):(4n+27). Find the ratio of their mth terms.

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Solution

Let a1, a2 be the first terms and d1, d2 the common differences of the two given A.P.'s .

Then the sums of their n terms are given by Sn = (n2) [2a1 + (n —1) d1], and

Sn = (n2) [2a2 + (n —1) d2]

SnSn = n2[2a1+(n1)d1]n2[2a2+(n1)d2]

= 2a1+(n1)d12a2+(n1)d2

It is given that

SnSn = 7n+14n+27 ... (i)

To find the ratio of the mth terms of the two given AP ' s, we replace n by (2m - 1) in equation (i).

Then we get

2a1+(2m2)d12a2+(2m2)d2 = 14m68m+23

Hence the ration of the mth terms of the two A.P 's is (14 m - 6) : (8 m + 23)


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