In an AP, if $m t_{m} = n t_{n}$ , then find $t_{m + n}$

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answer is D.

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Detailed Solution

$\begin{array}{l} m t_{m} = n t_{n} \\ \Rightarrow m (a + (m - 1) d) = n (a + (n - 1) d) \\ \Rightarrow (m - n) a = - (m^{2} - m - n^{2} + n) d \\ \Rightarrow (m - n) a = - [m^{2} - n^{2} - (m - n)] d \\ \Rightarrow a = - (m + n - 1) d \end{array}$

Now, $, t_{m + n} = a + (m + n - 1) d$

$\begin{aligned} = - (m + n - 1) d + (m + n - 1) d \\ = 0 . \end{aligned}$

Hence, the result

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