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Question

The sum of n terms of two A.Ps are in the ratio (5n+21):(3n13). The ratio of 24th terms is 1:?

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Solution

Let the two A.P's be a,a+d,a+2d,.... and a,a+d,a+2d,....
Let their sum be Sn and Sn respectively.
Sn=n2[2a+(n1)d] and Sn=n2[2a+(n1)d]
SnSn=3n135n+21
a+(n12)da+(n12)d=3n135n+21(1)
The 24th terms of two A.P's are respectively
T24=a+23d and T24=a+23d
T24T24=a+23da+23d(2)
Comparing (1) and (2), we get
n12=23
n=47
from (1),a+23da+23d=3×47135×47+21
=12
T24T24=12
T24:T24=1:2

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