# In an AP, if the ${m^{th}}$ term is n and ${n^{th}}$ term is m, then find the pth term. ($m \ne n$).

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Hint: Following the question we will get two equations. Subtracting them we will get the value of d and putting d value in one equation we will get the first term of the sequence. Substituting them in the general formula we will get the answer.

We know that in arithmetic progression, the general formula is,
${a_n} = a + (n - 1)d$
Where, ${a_n}$ = ${n^{th}}$ term of AP
a = first term of the AP
d = common difference in AP
Thus ${m^{th}}$ term, ${a_m} = a + (m - 1)d$
In the question it is given that ${m^{th}}$ term is n
i.e. $a + (m - 1)d = n$………………….(1)
And ${n^{th}}$ term, ${a_n} = a + (n - 1)d$
It is also given in the question that ${n^{th}}$ term is m
I.e. $a + (n - 1)d = m$………………..(2)
Subtracting equation (2) from equation (1) we find the common difference of the arithmetic series.
Hence, $[a + (m - 1)d] - [a + (n - 1)d] = n - m$
$\Rightarrow a + (m - 1)d - a - (n - 1)d = n - m$
Cancelling a and –a in left hand side we get,
$(m - 1)d - (n - 1)d = n - m$
Taking d common in left hand side we get,
$(m - 1 - n + 1)d = n - m$
Cancelling -1 and +1 we get,
$(m - n)d = n - m$
$\Rightarrow d = \dfrac{{n - m}}{{m - n}}$
Multiplying -1 on both side we get,
$d = - 1$
Putting value of d in equation (2) we get,
$a + (n - 1)d = m$
$\Rightarrow a + (n - 1)\left( { - 1} \right) = m$
$\Rightarrow a - n + 1 = m$
$\Rightarrow a = m + n - 1$
We got the value of a and d.
For the ${p^{th}}$ term,
We will use the general formula of AP i.e.
${a_n} = a + (n - 1)d$
Putting n = p, a = m+n-1 and d = -1 in the above formula we get,
${a_p} = \left( {m + n - 1} \right) + \left( {p - 1} \right) - 1$
Expanding the right hand side of the equation we get,
${a_p} = m + n - 1 - p + 1$
Cancelling -1 and +1 in the right hand side we get,
${a_p} = m + n - p$
Thus the ${p^{th}}$ term is ${a_p} = m + n - p$.

Note: Arithmetic progression or arithmetic sequence is the sequence in which the difference between two consecutive numbers are equal.
You can also subtract equation 1 from equation 2 to get the d value.
Be cautious while doing the equations because the mistakes in minus and plus signs can even change the whole answer.