The-first-term-of-a-sequence-is-1-the-second-is-2-and-every-term-is-the-sum-of-the-two-preceding-terms-The-n-th-term-is-

Question Number 25381 by Tinkutara last updated on 09/Dec/17

The first term of a sequence is 1, the

second is 2 and every term is the sum of

the two preceding terms . The n^{th} term

is .

Answered by prakash jain last updated on 09/Dec/17

a_{n} = a_{n - 1} + a_{n - 2}

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

x^{2} - x - 1 = 0

λ_{1} = \frac{1 + \sqrt{5}}{2}, λ_{2} = \frac{1 - \sqrt{5}}{2}

a_{n} = c_{1} {(\frac{1 + \sqrt{5}}{2})}^{n} + c_{2} {(\frac{1 - \sqrt{5}}{2})}^{n}

n = 0, a_{0} = 1

c_{1} + c_{2} = 1 (i)

n = 1, a_{1} = 2

c_{1} \frac{1 + \sqrt{5}}{2} + c_{2} \frac{1 - \sqrt{5}}{2} = 2

\sqrt{5} c_{1} = 2 - \frac{1 - \sqrt{5}}{2} \Rightarrow c_{1} = \frac{3 + \sqrt{5}}{2 \sqrt{5}}

c_{2} = 1 - c_{1} = \frac{\sqrt{5} - 3}{2 \sqrt{5}}

a_{n} = (\frac{3 + \sqrt{5}}{2 \sqrt{5}}) {(\frac{1 + \sqrt{5}}{2})}^{n} + (\frac{\sqrt{5} - 3}{2 \sqrt{5}}) {(\frac{1 - \sqrt{5}}{2})}^{n}

Commented by Tinkutara last updated on 09/Dec/17

W h a t i s c h a r a c t e r i s t i c e q u a t i o n ? A n d

w h y a_{n} = c_{1} λ_{1}^{n} + c_{2} λ_{2}^{n} ?

Commented by prakash jain last updated on 09/Dec/17

x^{2} - x - 1 = 0

λ_{1}^{2} - λ_{1} - 1 = 0

λ_{2}^{2} - λ_{2} - 1 = 0

c_{1} λ_{1}^{n - 2} (λ_{1}^{2} - λ_{1} - 1) + c_{2} λ_{2}^{n - 2} (λ_{2}^{2} - λ_{2} - 1) = 0

\Rightarrow (c_{1} λ_{1}^{n} + c_{2} λ_{2}^{n}) = (c_{1} λ_{1}^{n - 1} + c_{2} λ_{2}^{n - 1}) + (c_{1} λ_{1}^{n - 2} + c_{2} λ_{2}^{n - 2})

\Rightarrow a_{n} = a_{n - 1} + a_{n - 2}

The general solution of the equation

is a_{n} = c_{1} λ_{1}^{n} + c_{2} λ_{2}^{n}

This is the general solution when

λ_{1} \neq λ_{2}

when λ_{1} = λ_{2} = λ

a_{n} = (c_{1} + c_{2} n) λ^{n}

To know more

read linear difference equation

Commented by prakash jain last updated on 09/Dec/17

Characteristic equation is what

you get after replacing a_{n} by x^{n} .

Commented by Tinkutara last updated on 10/Dec/17

C a n w e a p p l y t h e s e m e t h o d s a n y w h e r e ?

F o r e x a m p l e, i n A G P ?