If (p + q)th and (p - q)th terms of a G.P. are m and n respectively,

1.	If (p + q)th and (p - q)th terms of a G.P. are m and n respectively, then write its pth term.
Answer» ⇒ Let the first term be a and the common ratio be r. ∴ According to the question, a_p+q = m. a_p-q = n. a_n = ar^n-1 a_p+q = a.r^p+q-1 a_p-q = a.r^p-q-1 ∴ a.r^p+q-1 = m. a.r^p-q-1 = n. Multiplying above two equations we get a²r^{(p+q-1+(p-q-1)} = a²r^(2p-2) a²r^(2p-2) = m.n (ar)^2(p-1) = m.n ∴ ar^p-1 =√m.n ⇒ P^th term is given by a.r^p-1 ∴ ar^p-1 =√m.n

If (p + q)th and (p - q)th terms of a G.P. are m and n respectively, then write its pth term.

Answer»

⇒ Let the first term be a and the common ratio be r.

∴ According to the question,

a_p+q = m.

a_p-q = n.

a_n = ar^n-1

a_p+q = a.r^p+q-1

a_p-q = a.r^p-q-1

∴ a.r^p+q-1 = m.

a.r^p-q-1 = n.

Multiplying above two equations we get

a²r^{(p+q-1+(p-q-1)} = a²r^(2p-2)

a²r^(2p-2) = m.n

(ar)^2(p-1) = m.n

∴ ar^p-1 =√m.n

⇒ P^th term is given by a.r^p-1

∴ ar^p-1 =√m.n