If m times the mth term of an ap is equal to n times its nth term

CBSE Class 11
1

If m times the mth term of an ap is equal to n times its nth term, show that (m+n)th term of ap is zero

2

Thge general nth term of an AP is a + (n -1)d.
From the given conditions,
m (a + (m-1)d) = n( a + (n-1)d)
⇒ am + m2d - md = an + n2d - nd
⇒ a(m-n) + (m+n)(m-n)d - (m-n)d = 0
⇒ (m-n) ( a + (m+n-1)d ) = 0
Rejecting the non-trivial case of m=n, we assume that m and n are different.
⇒ ( a + (m + n - 1)d ) = 0
The LHS of this equation denotes the (m+n)th term of the AP, which is Zero.