Consider two G.Ps. 2 , 2 2 , 2 3 , … and 4 , 4 2 , 4 3 , … of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is 2 225 8 , then ∑ k = 1 n k n - k is equal to:

Step-by-step solution

Given two G.Ps. 2 , 2 2 , 2 3 , … and 4 , 4 2 , 4 3 , … of 60 and n terms respectively, Also given the geometric mean of all the 60 + n terms is 2 225 8 , So, 2 1 2 2 … . 2 60 4 1 · 4 2 … … . 4 n 1 60 + n = 2 225 8 ⇒ 2 30 × 61 4 n n + 1 2 1 60 + n = 2 225 8 ⇒ 2 1830 + n 2 + n = 2 225 60 + n 8 On comparing both side we get, ⇒ 1830 + n 2 + n = 225 60 + n 8 ⇒ 8 n 2 - 217 n + 1140 = 0 ⇒ n = 20 , 57 8 Now ∑ k = 1 n k n - k = ∑ k = 1 20 n k