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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 mth terms.


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Solution

Find the ratio of m terms:

Step 1: Evaluate the sum for nth term

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+(n1)d1]

And,

Sn=n2[2a2+(n1)d2]

Therefore,

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

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

It is given that,

SnSn=7n+14n+272a1+(n1)d12a2+(n1)d2=7n+14n+27.................(1)

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

Therefore,

2a1+(2m2)d12a2+(2m2)d2=7(2m1)+14(2m1)+27a1+(m1)d1a2+(m1)d2=14m68m+23

Hence, the ratio of the mth terms of two A.P's is (14m6):(8m+23).


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