STEP III 2007 question 3 solution

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(i)

Fibonacci, yay!

(ii)

$F_{2k+3}F_{2k+1} - F_{2k+2}^2$

$= (F_{2k+2}+F_{2k+1})F_{2k+1} - F_{2k+2}^2$

$= F_{2k+2}F_{2k+1} +F_{2k+1}^2 - F_{2k+2}^2$

$= F_{2k+2}F_{2k+1} +F_{2k+1}^2 - F_{2k+2}(F_{2k+1}F_{2k})$

$= F_{2k+2}F_{2k+1} +F_{2k+1}^2 - F_{2k+2}F_{2k+1} - F_{2k+2}F_{2k}$

$= - F_{2k+2}F_{2k}+F_{2k+1}^2.$

(iii)

$P(n): \quad F_{2n+1}F_{2n-1} - F_{2n}^2 = 1$

Choosing n=2 gives:

$\text{LHS} = F_5F_3 - F_4^2 = 5\cdot 2 - 3^2 = 10 - 9 = 1 = \text{RHS} .$

Assume P(k):

$P(k) \Rightarrow F_{2k+1}F_{2k-1} - F_{2k}^2 = 1$

$\Rightarrow F_{2k+1}(F_{2k+1}-F_{2k}) - F_{2k}^2 = 1$

$\Rightarrow F_{2k+1}(F_{2k+1}-F_{2k}) - F_{2k}(F_{2k+2}-F_{2k+1}) = 1$

$\Rightarrow F_{2k+1}^2 - F_{2k+1}F_{2k} - F_{2k}F_{2k+2} + F_{2k}F_{2k+1} = 1$

$\Rightarrow F_{2k+1}^2 - F_{2k}F_{2k+2} = 1$

$\Rightarrow F_{2k+1}(F_{2k+3}-F_{2k+2}) - (F_{2k+2}-F_{2k+1})F_{2k+2} = 1$

$\Rightarrow F_{2k+1}F_{2k+3} - F_{2k+1}F_{2k+2} - F_{2k+2}^2+F_{2k+1}F_{2k+2} = 1$

$\Rightarrow F_{2k+1}F_{2k+3} - F_{2k+2}^2 = 1$

$\Rightarrow F_{2(k+1)+1}F_{2(k+1)-1} - F_{2(k+1)}^2 = 1 \Rightarrow P(k+1).$

$P(k) \Rightarrow P(k+1)$ which completes the induction.

Then $\displaystyle \frac{F_{2n}^2 + 1}{F_{2n+1}} = F_{2n-1}$ which is an integer.

(iv)

$F_{2n-1}^2 + 1$

$= (F_{2n+1} - F_{2n})^2 + 1$

$= (F_{2n+1} - F_{2n})^2 + (F_{2n+1}F_{2n-1} - F_{2n}^2)$

$= F_{2n+1}^2 - 2F_{2n+1}F_{2n} + F_{2n}^2 + F_{2n+1}F_{2n-1} - F_{2n}^2$

$= F_{2n+1}(F_{2n+1} - 2F_{2n} + F_{2n-1})$ as required.

Solution by generalebriety.

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