Question

If mth of AP is 1/n and nth term of AP is 1/m.show that matter is 1/2(mn+1)

Answer 1

Correct Question

If mth term of an A.P is 1/n and the nth term is 1/m show that the sum of mn term is ½(mn+1).

Solution

$\Longrightarrow{\mathsf{a_m = \frac{1}{n}}} \\ \\ \Longrightarrow{\mathsf{a_n = \frac{1}{m}}} \\ \\ \mathsf{On\:expanding\:at\:formula} \\ \\ \Longrightarrow{\mathsf{\frac{1}{n} = a + (m - 1)d ..(1)}} \\ \Longrightarrow{\mathsf{\frac{1}{m} = a + (n - 1)d ..(2)}}$

$\textsf{On doing (1) - (2) we get,}$

$\Longrightarrow{\mathsf{\frac{m - n}{nm} = d(m - n)}} \\ \Longrightarrow{\mathsf{d = \frac{1}{mn}}} \\ \\ \Longrightarrow{\mathsf{\frac{1}{m} = a + (n - 1)\frac{1}{mn}}} \\ \\ \Longrightarrow{\mathsf{\frac{1}{m} = a + \frac{1}{n} - \frac{1}{mn}}} \\ \\ \Longrightarrow{\mathsf{a = \frac{1}{mn}}}$

$\Longrightarrow{\mathsf{S_n = \frac{mn}{2} \times (2(\frac{1}{mn}) + (mn - 1)\frac{1}{mn})}} \\ \\ \Longrightarrow{\mathsf{S_n = \frac{mn}{2} \times (\frac{1}{mn} + 1)}} \\ \\ \Longrightarrow{\mathsf{S_n = \frac{1}{2} (mn + 1)}}$

Hence Proved