The sequence a_n _ n 0 is defined by a₀=1, a₁=-4 and a_ n+2 =-4 a_ n+1 -7 a_n , for n 0 . Find the number of positive integer divisors of a₅₀^2-a₄₉ a₅₁ .

Step-by-step solution

The sequence a_n , n 0 and a₀=1, a₁=-4 and given that a_ n+2 =-4 a_ n+1 -7 a_n ......(1) Now, aligned & a_n^2-a_ n+1 a_ n-1 =a_n^2- (4 a_n-7 a_ n-1 ) a_ n-1 & =a_n^2+4 a_n a_ n-1 +7 a_ n-1 ^2 & =-a_n (7 a_ n-2 )+7 a_ n-1 ^2 & =7 (a_ n-1 ^2-a_n a_ n-2 ) & =7^2 (a_ n-1 ^2-a_ n-1 a_ n-3 ) aligned ................................. array ll & a_n^2-a_ n+1 a_ n-1 =7^ n-1 (a₁^2-a₀ a₂ )=7^ n-1 (16-1 9)=7^n & a₅₀^2-a₄₉ a₅₁=7⁵⁰ array Number of positive integer divisors of 7⁵⁰=50+1=51