Question

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

Answer 1

Let a be the first term d be the common difference of the given AP. Then,
`T_(m) = a + (m-1)d "and" T_(n) = a + (n-1)d.`
Now, `(m * T_(m)) = (n * T_(n)) rArr m * {a+ (m-1)d} = n * {a +(n-1)d}`
`rArr a * (m-n) + {(m^(2) -n^(2)) - (m-n)} * d = 0`
`rArr (m-n) * {a + (m +n -1)}d.`
`rArr (m-n) * T_(m+n) = 0`
`rArr T_(m+n) = 0 [because (m-n) ne 0].`
Hence, the (m+n)th term is zero.