Question

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

Answer 1

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.