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

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

Explanation

Solution

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.

Complete step-by-step answer:
We know that in arithmetic progression, the general formula is,
an=a+(n1)d{a_n} = a + (n - 1)d
Where, an{a_n} = nth{n^{th}} term of AP
a = first term of the AP
d = common difference in AP
Thus mth{m^{th}} term, am=a+(m1)d{a_m} = a + (m - 1)d
In the question it is given that mth{m^{th}} term is n
i.e. a+(m1)d=na + (m - 1)d = n………………….(1)
And nth{n^{th}} term, an=a+(n1)d{a_n} = a + (n - 1)d
It is also given in the question that nth{n^{th}} term is m
I.e. a+(n1)d=ma + (n - 1)d = m………………..(2)
Subtracting equation (2) from equation (1) we find the common difference of the arithmetic series.
Hence, [a+(m1)d][a+(n1)d]=nm[a + (m - 1)d] - [a + (n - 1)d] = n - m
a+(m1)da(n1)d=nm\Rightarrow a + (m - 1)d - a - (n - 1)d = n - m
Cancelling a and –a in left hand side we get,
(m1)d(n1)d=nm(m - 1)d - (n - 1)d = n - m
Taking d common in left hand side we get,
(m1n+1)d=nm(m - 1 - n + 1)d = n - m
Cancelling -1 and +1 we get,
(mn)d=nm(m - n)d = n - m
d=nmmn\Rightarrow d = \dfrac{{n - m}}{{m - n}}
Multiplying -1 on both side we get,
d=1d = - 1
Putting value of d in equation (2) we get,
a+(n1)d=ma + (n - 1)d = m
a+(n1)(1)=m\Rightarrow a + (n - 1)\left( { - 1} \right) = m
an+1=m\Rightarrow a - n + 1 = m
a=m+n1\Rightarrow a = m + n - 1
We got the value of a and d.
For the pth{p^{th}} term,
We will use the general formula of AP i.e.
an=a+(n1)d{a_n} = a + (n - 1)d
Putting n = p, a = m+n-1 and d = -1 in the above formula we get,
ap=(m+n1)+(p1)1{a_p} = \left( {m + n - 1} \right) + \left( {p - 1} \right) - 1
Expanding the right hand side of the equation we get,
ap=m+n1p+1{a_p} = m + n - 1 - p + 1
Cancelling -1 and +1 in the right hand side we get,
ap=m+np{a_p} = m + n - p
Thus the pth{p^{th}} term is ap=m+np{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.