In an A.P, if $m^{th}$ term is n and the $n^{th}$ term is m, where $m \neq n$, find the $p^{th}$ term.

In an A.P., the $m^{th}$ term is $n$ and the $n^{th}$ term is $m$.

For any term in an Arithmetic Progression:

$T_k = a + (k - 1)d$

where $a$ is the first term, $d$ is the common difference, and $k$ is the position of the term.

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Using the given information:

For the $m^{th}$ term:

$a + (m - 1)d = n$ ... (1)

For the $n^{th}$ term:

$a + (n - 1)d = m$ ... (2)

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Subtracting equation (2) from equation (1):

$[a + (m - 1)d] - [a + (n - 1)d] = n - m$

The $a$ cancels out:

$(m - 1)d - (n - 1)d = n - m$

Factoring out $d$:

$d(m - 1 - n + 1) = n - m$

$d(m - n) = n - m$

Since $n - m = -(m - n)$:

$d(m - n) = -(m - n)$

Dividing both sides by $(m - n)$ (we can do this because $m \neq n$):

$d = -1$

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Substituting $d = -1$ into equation (1):

$a + (m - 1)(-1) = n$

$a - (m - 1) = n$

$a - m + 1 = n$

$a = n + m - 1$

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Using the A.P. formula for the $p^{th}$ term:

$T_p = a + (p - 1)d$

Substituting $a = n + m - 1$ and $d = -1$:

$T_p = (n + m - 1) + (p - 1)(-1)$

$T_p = n + m - 1 - (p - 1)$

$T_p = n + m - 1 - p + 1$

$T_p = n + m - p$

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Therefore, the $p^{th}$ term is $n + m - p$.