Find the 15 t h term of the series 3 + 1 3 + 1 3 3 + … .

Given series is 3 + 1 3 + 1 3 3 + ⋯ ⋯ This is a GP with a 1 = 3 and common ratio r = a 2 a 1 = 1 3 × 1 3 = 1 3 n t h term in a GP is a n = a 1 r n - 1 , where a 1 is first term and r is common ratio. ∴ a 15 = a r 14 = 3 1 3 14 ∴ 15 t h term of given GP is 3 1 3 14 .

Find the n t h term of the series 1 2 - 2 + 4 2 - 16 + … .

Given sequence is 1 2 - 2 + 4 2 - 16 + ⋯ i.e., 1 2 , - 2 , 4 2 , - 16 , ⋯ This is a GP with a 1 = 1 2 and common ratio r = a 2 a 1 = - 2 × 2 = - 2 2 n t h term in a GP is a n = a 1 r n - 1 , where a 1 is first term and r is common ratio. ∴ a n = a r n - 1 = 1 2 ( - 2 2 ) n - 1 = - 1 n - 1 2 2 n - 1 ( 2 ) n - 1 = - 1 n - 1 · 2 n - 1 · ( 2 ) n - 2 ∴ n t h term of given GP is - 1 n - 1 · 2 n - 1 · ( 2 ) n - 2 .

Find the 6 t h and the n t h terms of the sequence 3 2 , 3 4 , 3 8 , … .

Given sequence is 3 2 , 3 4 , 3 8 , ⋯ This is a GP with a 1 = 3 2 and common ratio r = a 2 a 1 = 3 4 × 2 3 = 1 2 n t h term in a GP is a n = a 1 r n - 1 , where a 1 is first term and r is common ratio. ∴ a 6 = a r 5 = 3 2 1 2 5 = 3 2 6 ∴ 6 t h term of given GP is 3 64 . a n = a r n - 1 ⇒ a n = 3 2 1 2 n - 1 ∴ n t h term of a given GP is 3 2 n .