<s>[INST] on a recursive sequence (exercise 8.14 Apostol). The exercise asks to prove convergence and find the limit of the sequence:$$a_n= \frac{b_{n+1}}{b_n},\text{ where } b_1=b_2 =1, b_{n+2} = b_{n} + b_{n+1}. $$
It also gives a hint: Show that  $ \ b_{n+2}b_n - b_{n+1}^2 = (-1)^{n+1}$ by induction.
I am having problems proving $ \ b_{n+2}b_n - b_{n+1}^2 = (-1)^{n+1}$ by induction, could I have a helping hand?
 [/INST] $$b_{n+3}b_{n+1}-b_{n+2}^2= (b_{n+2}+b_{n+1})b_{n+1}-b_{n+2}^2 = (b_{n+1}-b_{n+2})b_{n+2} + b^2_{n+1} = (b_{n+1}-b_{n+2})(b_{n+1} + b_n) + b^2_{n+1}= 2b^2_{n+1}-b_{n+2}b_{n+1} +b_{n+1}b_n -b_{n+2}b_n = (b^2_{n+1}-b_{n+2}b_n)+b^2_{n+1}-b_{n+2}b_{n+1} +b_{n+1}b_n$$
Now note that $$b^2_{n+1}-b_{n+2}b_{n+1} +b_{n+1}b_n = b_{n+1}(b_{n+1}+b_n-b_{n+2}) = 0$$
and therefore you get that $$b_{n+3}b_{n+1}-b_{n+2}^2= (b^2_{n+1}-b_{n+2}b_n)+b^2_{n+1}-b_{n+2}b_{n+1} +b_{n+1}b_n = -(b_{n+2}b_n-b^2_{n+1}) = -(-1)^{n+1} = (-1)^{n+2}$$
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