If m times the mth term of an A.P. is equal to n times its nth term, t

1.	If m times the mth term of an A.P. is equal to n times its nth term, then show that the (m + n)th term of the A.P. is zero.
Answer» Given, mt_m = nt_n m[a + (m – 1) d] = n [a + (n – 1) d] [we know that t_n = a + (n – 1)d] m[a + md – d] = n[a + nd – d] ma + m²d – md = na + n²d – nd ma – na + m²d – n²d = md – nd a (m – n) + d (m² – n²) = d(m – n) a (m – n) + d(m + n)(m – n) = d(m – n) ÷ by (m – n) on both sides, a + d (m + n) = d a + d(m + n) – d = 0 a + d(m + n – 1) = 0 … (1) To prove, t_{m + n} = 0 t_{m + n} = a + (m + n – 1)d t_{m + n} = 0 (from(1)) Hence it is proved.

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

Answer»

Given, mt_m = nt_n

m[a + (m – 1) d] = n [a + (n – 1) d]

[we know that t_n = a + (n – 1)d]

m[a + md – d] = n[a + nd – d]

ma + m²d – md = na + n²d – nd

ma – na + m²d – n²d = md – nd

a (m – n) + d (m² – n²) = d(m – n)

a (m – n) + d(m + n)(m – n) = d(m – n) ÷ by (m – n) on both sides,

a + d (m + n) = d

a + d(m + n) – d = 0

a + d(m + n – 1) = 0 … (1)

To prove, t_{m + n} = 0

t_{m + n} = a + (m + n – 1)d

t_{m + n} = 0 (from(1))

Hence it is proved.