Really just need help with part c)

My estimate of the limit is .618

Look at part d of the link I'm going to post its talking about shifted sequences. Your limits should both be L or in your case .618 for a_n and a_n+1
http://www.math.uiuc.edu/~hildebr/347.summer14/epsilonics1sol.pdf

if L is the limit, and assuming L is not -1 (this part is true because every a_n is obviously a positive real), then :

L = 1 / ( 1 + L )

since L is the common limit of sequences (a_n) and (a_n+1).

solve for L : discard the negative solution, and you'll get the answer L = ( -1 + sqrt(5) ) / 2

on a sidenote, this limit is the reciprocal of the golden number !

Can you explain this a bit more, I'm a bit lost

You have may a theorem in your book

If an converges to L then every subsequence of an converges to L

If you have this the substitute for L and solve

Depending on how much your instructor wants you may want to show its a positive real by monotonicty and boundedness

Can you explain how you reached (-1+sqrt(5))/2 by solving?

actually quite simple.

L = 1 / ( 1 + L )

L(1+L) = 1
L^2 + L -1 = 0
ue quadratic eqn

[-1+/- sqr(1-(-4))]/2

L= [-1 + sqrt(5)]/2
leave it up to u why -sqrt isnt used

Thanks can't believe I didn't see that before