Question
Question: If U<sub>n</sub> = (\(\sqrt{3}\)+ 1)<sup>2n</sup> + (\(\sqrt{3}\)– 1)<sup>2n</sup> then-...
If Un = (3+ 1)2n + (3– 1)2n then-
A
4Un+1 = 2Un – 3Un–1
B
Un+1 = 4Un – Un–1
C
Un+1 = 8Un – 4Un–1
D
Un+1 = 2Un
Answer
Un+1 = 8Un – 4Un–1
Explanation
Solution
Un = [(3 + 1)2]n + [(3– 1)2]n
= (4 + 23)n + (4 – 23)n
= an + bn where a + b = 8, ab = 4
8Un = (a + b) (an + bn)
= an+1 + bn+1 + abn + ban
= Un+1 + ab (an–1 + bn–1)
= Un+1 + 4Un–1
Ž Un+1 = 8Un – 4Un–1.