Question

if m times the mth term is equal to n times the nth term. then what is (m+n)th term of an a.p

Answer 1

Dear Student!

If the question is “if n times the m^th term is equal to m times the n^th term, then what is (m + n)^th term of the A.P.”, then the answer is given below.

Let the first term and common difference of the A.P. be ‘a’ and ‘d’ respectively.

Then, m^th term = a + (m – 1) d and n^th term = a + (n – 1) d

By the given condition,

n [a + (m – 1) d] = m [a + (n – 1) d]

⇒ na + n (m – 1) d = ma + m (n – 1) d

⇒ n(m – 1)d – m (n – 1)d = ma – na

⇒ d[n(m – 1) – m (n – 1)] = a (m – n)

⇒ d[nm – n – mn + m] = a (m – n)

⇒ d(m – n) = a (m – n)

⇒ d = a

Now (m + n)^th term = a + (m + n – 1)d.

But d = a

∴ (m + n)^th term = a + (m + n – 1) a = a + a(m + n) – a = a (m + n)

Best wishes!