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Question

If m times the mth term is equal to n times the nth term of an A.P. Prove that (m+n)thterm of A.P is zero


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Solution

Step 1: Find the relation between first term and common difference of given A.P.

A.P is a sequence of numbers in order in which difference between two consecutive numbers remains constant that is known as common difference.

Let the first term of the given A.P. be a and its common difference be d.

As the nth term of an A.P. is an=a+n-1d

So, its mth term will be am=a+m-1d

According to the question, mam=nan

ma+m-1d=na+n-1d

ma+mm-1d=na+nn-1d

ma-na=nn-1d-mm-1d

m-na=n2-nd-m2+md

m-na=-m2-n2d+m-nd

m-na=-m-nm+nd+m-nd [a2-b2=a-ba+b]

a=-m+nd+d

a=-m+n-1d

Step 2: Proving that (m+n)thterm of given A.P is zero.

Now, (m+n)th term of A.P. is

am+n=a+m+n-1d

am+n=-m+n-1d+m+n-1d [From Step 1]

am+n=0

Hence, it is proved that (m+n)thterm of A.P is zero.


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