The nth term of an A.P. is n and its nth term is m. Prove that its pth term is $m + n - p$ . Also show that its $(m + n)$ th term is zero.

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Solution

$T_{m} = a + (m - 1) d = n, T_{n} = a (n - 1) d = m$
Solving, we get
$d = - 1$ and $a = m + n - 1$ .
$∴ T_{p} = a + (p - 1) d$
$= m + n - 1 + (p - 1) (- 1) = m + n - p$ .
$T_{m + n} = a + (m + n - 1) d$
$= (m + n - 1) + (m + n - 1) (- 1) = 0$ .

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