Fitzpatrick Polynomial Q (1 Viewer)

[Roobs]

hey...ive got a question here from 36c fitzpatrick:

q13:

if alpha, beta are the roost of x^2-px+q=0
Sn denotex (alpha)^n +(beta)^n, where n is a positive integer, show that:

Sn+2 - pSn+1 + qSn = 0

hence or otherwise find S3, S4, S5 in terms of p and q. Show by this method that (SQRT 2 + 1)^10 + (SQRT 2 - 1)^10=6726

thanks for any and all replies

 

[Raginsheep]

Since a and b are the roots of the quadratic, you can write (x-a)(x-b) and equate coefficients to get p=a+b and q=ab.

 

[Yip]

Substitute alpha into quadratic and multiply by alpha^n
Substitute beta into quadratic and multiply by beta^n
Add 2 equations to get required recursion

sub n=3 into recursion and reduce until u get to s0 to find s3,s4 and s5

Then, find the quadratic with sqrt2 +1 and sqrt2-1 as roots (x^2-2sqrtx+1)
and find S10 using the recursion given previously i think

 

[Roobs]

Yip you genious...the first part fell out in 5 lines---i had 2 attempts at 3 pages without a solution........"bangs head against wall"...im working on part 2 now....