if-a-1-3-a-2-2-a-n-2-a-n-1-a-1-2-find-a-6-mister-W-method-a-n-A-1-3-2-n-B-1-3-2-n-a-1-A-1-3-2-B-1-3-2-3-a-2-A-1-3-2-2-B-1-3-

Question Number 81458 by jagoll last updated on 13/Feb/20

if a_{1} = 3, a_{2} = 2

a_{n + 2} = a_{n + 1} + \frac{a_{1}}{2}

find a_{6} = ?

mister W method

a_{n} = A {(\frac{1 + \sqrt{3}}{2})}^{n} + B {(\frac{1 - \sqrt{3}}{2})}^{n}

a_{1} = A (\frac{1 + \sqrt{3}}{2}) + B (\frac{1 - \sqrt{3}}{2}) = 3

a_{2} = A {(\frac{1 + \sqrt{3}}{2})}^{2} + B {(\frac{1 - \sqrt{3}}{2})}^{2} = 2

\Rightarrow A + B + (A - B) \sqrt{3} = 6

\Rightarrow 2 (A + B) + (A - B) \sqrt{3} = 4

A = \frac{4 - \sqrt{3}}{\sqrt{3}}, B = \frac{- 4 - \sqrt{3}}{3}

a_{n} = (\frac{4 - \sqrt{3}}{\sqrt{3}}) {(\frac{1 + \sqrt{3}}{2})}^{n} - (\frac{4 + \sqrt{3}}{\sqrt{3}}) {(\frac{1 - \sqrt{3}}{2})}^{n}

Commented by jagoll last updated on 13/Feb/20

mister W, my work is correct ?

Commented by john santu last updated on 13/Feb/20

yes