The ratio of the sums of $m$ and $n$ terms of an A.P is $m^{2} : n^{2},$ if the ratio of the $m^{t h}$ and $n^{t h}$ terms is $(2 m - 1) : (2 n - 1)$ , write $1$ else $0.$

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Solution

$S_{m} : S_{n} = m^{2} : n^{2}$
$\Rightarrow \frac{\frac{m}{2} [2 a + (m - 1) d]}{\frac{n}{2} [2 a + (n - a) d]} = \frac{m^{2}}{n^{2}}$
$\Rightarrow \frac{2 a + (m - 1) d}{2 a + (n - a) d} = \frac{m}{n}$
$\Rightarrow 2 a n + n (m - 1) d = 2 a m + m (n - 1) d$
$\Rightarrow 2 a (n - m) + d [n m - n - m n + m] = 0$
$\Rightarrow 2 a (n - m) = d (n - m)$
$\Rightarrow 2 a = d$
So, $\frac{a_{m}}{a_{n}} = \frac{a + (m - 1) d}{a + (n - 1) d}$
$\frac{a_{m}}{a_{n}} = \frac{a + (m - 1) 2 a}{a + (n - 1) 2 a}$
$= \frac{a [1 + (m - 1) 2]}{a [1 + (n - 1) 2]}$
$= \frac{2 m - 1}{2 n - 1}$

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The ratio of the sums of m and n terms of an A.P is m2:n2, if the ratio of the mth and nth terms is (2m−1):(2n−1), write 1 else 0.

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