Question
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 term is n and nth term is m, then find the pth term. (m=n).
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+(n−1)d
Where, an = nth term of AP
a = first term of the AP
d = common difference in AP
Thus mth term, am=a+(m−1)d
In the question it is given that mth term is n
i.e. a+(m−1)d=n………………….(1)
And nth term, an=a+(n−1)d
It is also given in the question that nth 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
⇒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
⇒d=m−nn−m
Multiplying -1 on both side we get,
d=−1
Putting value of d in equation (2) we get,
a+(n−1)d=m
⇒a+(n−1)(−1)=m
⇒a−n+1=m
⇒a=m+n−1
We got the value of a and d.
For the pth term,
We will use the general formula of AP i.e.
an=a+(n−1)d
Putting n = p, a = m+n-1 and d = -1 in the above formula we get,
ap=(m+n−1)+(p−1)−1
Expanding the right hand side of the equation we get,
ap=m+n−1−p+1
Cancelling -1 and +1 in the right hand side we get,
ap=m+n−p
Thus the pth term is ap=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.