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Question Number 54341 by Meritguide1234 last updated on 02/Feb/19

Answered by iv@0uja last updated on 03/Feb/19

$$\underset{t=0}{\sum ^s}{}_s{C}_t{\left(\frac{\sqrt{5}-1}{2}\right)}^t={(1+\frac{\sqrt{5}-1}{2})}^s={\left(\frac{\sqrt{5}+1}{2}\right)}^s$$$$\underset{s=0}{\sum ^r}{}_r{C}_s{\left(\frac{2}{1+\sqrt{5}}\right)}^s={(1+\frac{2}{1+\sqrt{5}})}^r={\left(\frac{3+\sqrt{5}}{1+\sqrt{5}}\right)}^r$$$$\underset{r=0}{\sum ^n}{}_n{C}_r{\left(\frac{1+\sqrt{5}}{3+\sqrt{5}}\right)}^r={(1+\frac{1+\sqrt{5}}{3+\sqrt{5}})}^n={\left(\frac{4+2\sqrt{5}}{3+\sqrt{5}}\right)}^n$$$$\sqrt[n]{(...)}=\frac{4+2\sqrt{5}}{3+\sqrt{5}}=\frac{1+\sqrt{5}}{2}$$

Commented by Meritguide1234 last updated on 03/Feb/19

$$\mathrm{nice}$$

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