Let a₁, a₂, , a_n be the roots of the polynomial _ k=1 ^n k x^k=0 . If _ k=1 ^n 1 (1-a_k )^2 =-13 . Find n .

Step-by-step solution

Let aligned P(x) & = _ k=1 ^n k x^k n (x-a₁ ) (x-a₂ ) (x-a_n ) P(x) & = n+ _ i=1 ^n (x-a_i ) aligned Differentiate two times and put x=1 , we get P(1) P^ (1)- (P^ (1) )^2 (P(1))^2 = _ k=1 ^n 1 (1-a_k )^2 =13 Now, P(x)= _ k=1 ^n k x^k aligned P(1) & = _ k=1 ^n k= n(n+1) 2 P^ (1) & = _ k=1 ^n k^2= n(n+1)(2 n+1) 6 P^ (1)= _ k=1 ^n k^2(k-1) & = n^3- n^2= ( n(n+1) 2 )^2- n(n+1)(2 n+1) 6 aligned On putting these values we get n=22 .