Question

If m times the m^{​​​​​​th} term of an AP is equal to n times the n^{​​​​​​th} term, find its (m+n)^th term.

Answer 1

Given m × t_m = n × t_n
⇒ m[a + (m – 1)d] = n[a + (n – 1)d]
⇒ ma + m(m – 1)d] = na + n(n – 1)d
⇒ ma + m(m – 1)d] – na – n(n – 1)d = 0
⇒ a(m – n) + [m² – m– n² + n]d = 0
⇒ a(m – n) + [m² – n² – m + n]d = 0
⇒ a(m – n) + [(m – n)(m + n) – (m – n)]d = 0
⇒ (m – n)[a+ {(m + n) – 1}]d = 0
⇒ [a+ {(m + n) – 1}]d = 0
Hence t_{(m + n)} = 0