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Question

If mtimes the mthterm of an A.P. is equal to ntimes the nthterm, then (m+n)thterm of the A.P is _______ .


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Solution

According to nth term of an arithmetic progression, we have

tn=nthterm of AP =a+n-1d............(1)

tm=mthterm of AP =a+m-1d............(2)

From the given information, we can write

m×tm=n×tn

From (1)and(2) we can write,

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

Now, simplify the equation, we get

am+m2d-md=an+n2d-nda(m-n)+(m+n)(m-n)d-(m-n)d=0

Now, take out common terms

(m-n)[a+(m+n-1)d]=0

(a+(m+n-1)d)=0.......(3)

From equation (3),we can say

(m-n)[a+(m+n-1)d]=tm+n......(4)

Now comparing equations (3)and(4), we can conclude that

tm+n=0

Hence, (m+n)thterm of A.P. is 0.


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