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Question

If the sums of n terms of two arithmetic progressions are in the ratio 2n+5:3n+4, then write the ratio of their mth terms.

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Solution

It is given that

SnS1n=2n+53n+4

Let Sn=n2[2a1+(n1)d1]

and, S1n=n2[2a2+(n1)d2]

SnSn=n2[2a1+(n1)d1]n2[2a2+(n1)d2]=2n+53n+4

2a1+(n1)d12a2+(n1)d2=2n+53n+4

a1+(n12)d1a2+(n12)d2=2n+53n+4

Let, n12=m

n1=2m

n=2m1

Replacing n12 by m on both side of equation (ii), we get

a1+md1a2+md2=2(2m1)+53(2m1)+4

=4m2+56m3+4

=4m+36m+1

Ratio of the mth terms =4m+36m+1


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