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

1.	If m times the mth term of an A.P. is equal to n times nth term, then show that the (m + n)th term of the A.P. is zero
Answer» According to the given condition, mt_m = nt_n ∴ m[a + (m – 1)d] = n[a + (n – 1)d] ∴ ma + md(m – 1) = na + nd(n- 1) ∴ ma + m²d – md = na + n²d – nd ∴ ma + m²d – md – na – n²d + nd = 0 ∴ (ma – na) + (m²d – n²d) – (md – nd) = 0 ∴ a(m – n) + d(m² – n²) – d(m – n) = 0 ∴ a(m – n) + d(m + n) (m – n) – d(m – n) = 0 ∴ (m – n)[a + (m + n – 1) d] = 0 ∴ [a+ (m + n – 1)d] = 0 …[Dividing both sides by (m – n)] ∴ t_{(m + n)} = 0 ∴ The (m + n)^th term of the A.P. is zero.

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

Answer»

According to the given condition,

mt_m = nt_n

∴ m[a + (m – 1)d] = n[a + (n – 1)d]

∴ ma + md(m – 1) = na + nd(n- 1)

∴ ma + m²d – md = na + n²d – nd

∴ ma + m²d – md – na – n²d + nd = 0

∴ (ma – na) + (m²d – n²d) – (md – nd) = 0

∴ a(m – n) + d(m² – n²) – d(m – n) = 0

∴ a(m – n) + d(m + n) (m – n) – d(m – n) = 0

∴ (m – n)[a + (m + n – 1) d] = 0

∴ [a+ (m + n – 1)d] = 0 …[Dividing both sides by (m – n)]

∴ t_{(m + n)} = 0

∴ The (m + n)^th term of the A.P. is zero.