Paragraph: Let a_n and b_n be the arithmetic sequences each with common difference 2 such that a₁ b₁ and let c_n= _ k=1 ^n a_k, d_n= _ k=1 ^n b_k . Suppose that the points A_n (a_n, c_n ), B_n (b_n, d_n ) are all lies on the parabola C: y=p x^2+q x+r where p, q, r are the constants. Question: The value of p equals

Step-by-step solution

Given c_n=a₁+a₂+a₃+ . .+a_n where a₁, a₂, . ., a_n are in A.P. with d=2 and d_n=b₁+b₂+b₃+ +b_n where b₁, b₂, b₃, b_n are in A.P. with d=2 Also (a_n, c_n ) lies on y=p x^2+q x+r Now c_n=p a_n^2+q a_n+r ....(1) c_ n-1 =p a_ n-1 ^2+q a_ n-1 +r ....(2) From (1) and (2), we get c_n-c_ n-1 =p (a_n^2-a_ n-1 ^2 )+q (a_n-a_ n-1 ) a_n=p (a_n+a_ n-1 ) (a_n-a_ n-1 )+q (a_n-a_ n-1 )a_n= (a_n-a_ n-1 ) [p (a_n+a_ n-1 )+q ] ....(3) (a_n-a_ n-1 =d )