If m times the m^th term is equal to n times the n^th term of an A.P. prove that (m+n)^th term of A.P is zero.

An arithmetic progression is a sequence where the differences between every two consecutive terms are the same.

Answer: We have proved that, if m times the m^th term is equal to n times the n^th term of an A.P., (m+n)^th term of A.P is equal to zero.

Let's look into the explanation.

Explanation:

We know that, according to the n^th term of an AP we have,

n^th term of AP = tₙ = a + (n − 1)d ------------- (1)

m^th term of AP = tₘ​ = a + (m − 1)d ------------- (2)

According to the question, m times the m^th term is equal to n times the n^th term which can be written as,

mtₘ​ = ntₙ

From equation (1) and (2),

m[a + (m − 1)d] = n[a + (n − 1)d]

m[a + (m − 1)d] − n[a + (n − 1)d] = 0

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

(m − n)[a + d((m + n) − 1)] = 0

a + [(m + n) − 1]d = 0 ------------- (3)

But, a + [(m + n) − 1]d = tₘ ₊ ₙ

Thus, tₘ ₊ ₙ = 0 [From equation (3)]

Hence we have proved that, if m times the m^th term is equal to n times the n^th term of an A.P., (m+n)^th term of A.P is equal to zero.