Given-the-series-a-n-3a-n-1-2a-n-2-with-a-1-11-amp-a-2-21-Find-a-n-

Question Number 137899 by bemath last updated on 08/Apr/21

G i v e n t h e s e r i e s

a_{n} = 3 a_{n - 1} + 2 a_{n - 2} w i t h

a_{1} = 11 & a_{2} = 21 . F i n d a_{n} .

Answered by benjo_mathlover last updated on 08/Apr/21

c h a r a c t e r i s t i c s e q u a t i o n

r^{2} - 3 r - 2 = 0; r = \frac{3 \pm \sqrt{17}}{2}

a_{n} = A . {(\frac{3 + \sqrt{17}}{2})}^{n} + B {(\frac{3 - \sqrt{17}}{2})}^{n}

w i t h n = 1 \Rightarrow 11 = \frac{3}{2} (A + B) + \frac{\sqrt{17}}{2} (A - B)

w i t h n = 2 \Rightarrow 21 = \frac{13}{2} (A + B) + \frac{3 \sqrt{17}}{2} (A - B)

A = (\frac{20 - 3 \sqrt{17}}{\sqrt{17}}); B = - (\frac{20 + 3 \sqrt{17}}{\sqrt{17}})

∴ a_{n} = (\frac{20 - 3 \sqrt{17}}{\sqrt{17}}) {(\frac{3 + \sqrt{17}}{2})}^{n} - (\frac{20 + 3 \sqrt{17}}{\sqrt{17}}) {(\frac{3 - \sqrt{17}}{2})}^{n}