In an arithmetic progression the p + 1 t h term is twice the q + 1 t h term. If its 3 p + 1 t h term is λ times the p + q + 1 t h term, then λ is equal to

Step-by-step solution

For the A.P. let a be the first term and d be the common difference Given T p + 1 = 2 T q + 1 ⇒ a + p d = 2 a + q d ⇒ a = p - 2 q d Also, T 3 p + 1 = λ T p + q + 1 a + 3 p d = λ a + p + q d a λ - 1 = d 3 p - λ p - λ q So, p - 2 q d = λ p + λ q - 3 p d 1 - λ ⇒ p - p λ - 2 q + 2 q λ = λ p + λ q - 3 p ⇒ λ p + q + p - 2 q = p - 2 q + 3 p ⇒ λ = 4 p - 2 q 2 p - q = 2 2 p - q 2 p - q = 2