If m times the mth term of an A.P. is equal to n times nth term, then (m+n)th term of the A.P is ____.

If m times the mth term of an A.P. is equal to n times nth term, then (m+n)th term of the A.P is zero.

Explanation:

According to the nth term of an arithmetic progression, we have

tn = nth term of Ap = a + (n − 1)d … (1)

tm = mth term of AP = a + (m − 1)d … (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 − 1)d]

Now, simplify the equation, we get

m[a + (m − 1)d] − n[a + (n − 1)d] = 0

a(m − n) + d[(m + n)(m − n) − (m − n)] = 0

Now, take out the common terms, we get

(m − n)[a + d((m + n) − 1)] = 0

a + [(m + n) − 1]d = 0 … (3)

From equation (3), we can say

a + [(m + n) − 1]d = tm+n …. (4)

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

tm+n = 0.

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