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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 26
Total: 106
The ratio of the sums of m and n terms of an A.P. is m2:n2. Show that the ratio of mth and nth term is (2m – 1) : (2n – 1).
Solution:  
Let the first term be a & common difference be d. Then
Sm
sn
 =
m2
n2
 ⇒
m
2
[2a+(m−1)d]
n
2
[2a+(n−1)d]
 =
m2
n2
 ⇒
2a+(m−1)d
2a+(n−1)d
 =
m
n
⇒ 2an + (m – 1)nd = 2am + (n – 1)md
⇒ 2a (n – m) = (nm – m – mn + n)d ⇒ 2a = d ....(i)
Now required ratio,
a+(m−1)d
a+(n−1)d
 =
a+(m−1).2a
a+(n−1).2a
 [by (i)]
=
1+2m−2
1+2n−2
 =
2m−1
2n−1
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