Question:
If m times the mth term of an A.P. is equal to n times its nth term, show that (m n)th term of the A.P. is zero.
Answer:
The nth term of an AP is given as: a + (n -1)d
From the given conditions,
m*mth term= n*nth term
=> m (a + (m-1)d) = n( a + (n-1)d)
=> am + m2d - md = an + n2d - nd
=> am + m2d - md - an - n2d + nd = 0
=> a(m-n) + (m+n)(m-n)d - (m-n)d = 0
=> (m-n)(a + (m+n-1)d ) = 0
=> 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
=> (m+n)th term = 0
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