Paragraph: Two AP's having same number of terms equal to k . The ratio of the last term of the first progression to the first term of the second progression equals the ratio of the last term of the second progression to the first term of the first progression both of which are numerically equal to 4 . The ratio of the sum of k terms of the first progression to the sum of k terms of second progression is equal to 2 .

Step-by-step solution

Let the first A.P. be a₁, a₂, a₃, . ., a_k and the second A.P. be b₁, b₂, b₃, , b_k Given a_k b₁ = b_k a₁ =4 array lll a_k=4 b₁ & & a₁+(k-1) d₁=4 b₁ ...(1) b_k=4 a₁ & & b₁+(k-1) d₂=4 a₁ ... (2) array Given S_k S_k^ =2 2 a₁+(k-1) d₁ 2 b₁+(k-1) d₂ =2 aligned & a₁+ a₁+(k-1) d₁ ^ b₁+(k-1) d₂ b₁ =2 & a₁+4 b₁ b₁+4 a₁ =2 a₁+4 b₁=2 b₁+8 a₁ aligned a₁ b₁ = 2 7 ... (3) From eqns. (1) and (2) aligned & (k-1) d₁=4 b₁-a₁ & (k-1) d₂=4 a₁-b₁ aligned On dividing, we get d₁ d₂ = 4 b₁-a₁ 4 a₁-b₁ = 4-(2 / 7) 4(2 / 7)-1 =26= Also arra