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 lying on the parabola C : y = px ^2+q x+r where p, q, r are constants. Question: The value of q equals

Step-by-step solution

(i) Given c _ n = a ₁+ a ₂+ a ₃+ .+ 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 = pa _ n ^2+ qa _ 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 ) On putting n=2 and 3 in equation (3), we get a₂=d [