Maths help needed- sequences and series, 10pts!

For the purpose of this question ^ is to the power of.

A sequence of terms (U^k) is defined by k as greater than or equal to 1 by the recurrence relation U^k+2 = U^k+1 - pU^k, where p is a constant. Given that u^1=2 and U^2 =4:

a) find an expression in terms of p for U^3.
b) Hence find an expression in terms of p for u^4
c) Given that U^4 is twice the value of U^3, find the value of P.

could anyone please answer this and show their full method, I am really struggling on recurrence relations.

thanks in advance

This is a geometric series with U as the ratio. Since

u^1 = 2 &
u^2 = 4, we can see that u = 2. Hence, the ratio is 2.

U^(k+1) - pU^k = U^(k+2)
U^k(U - p) = U^k * U^2
U - p = U^2
U - U^2 = p
2 - 4 = p
-2 = p

a)
(U - U^2)^3 = p^3

b)
(U - U^2)^4 = p^4

c)
We know that U = 2, therefore we have:

p = -2


Hope this helped.

a) U^3 = U^2 - pU^1 = 4 - 2p
b) U^4 = U^3 - pU^2 = (4 - 2p) - 4p = 4 - 6p
c) 4 - 6p = 2 * (4 - 2p) = 8 - 4p
-4 = 2p
p = -2