For n 1 2 10 show that 5un2 41n is always a perfect square

For n = 1, 2,..., 10, show that 5u_n^2 + 4(-1)^n is always a perfect square.

We can prove it using the principle of mathematical Induction

Base Case (n=1)

Number = 5u1^2 + 4(-1)^1 = 5(1) + 4(-1) = 1, which is a perfect square

Assumption Step(n=9)

Let us assume that the given thing holds for n=9

5u9^2 + 4(-1)^9 is a perfect square

Induction Step(n=10)

5u10^2 + 4(-1)^10

=> 5(u9 + u8)^2 + 4(-1)^10

=> 5(u9^2 + u8^2 + 2u8u9) + 4(-1)^10

which is also a perfect square

Hence by using the principle of mathematical induction, the given statement holds true