#1804 · Number Theory · Diophantine equations

Convergents of the Golden Ratio

If $\frac{p _{r}}{q _{r}}$ denote the $r^{th}$ convergent to $\frac{5 + 1}{2}$ , show that $p_{3} + p_{5} + \dots + p_{2 n - 1} = p_{2 n} - p_{2}$ and $q_{3} + q_{5} + \dots + q_{2 n - 1} = q_{2 n} - q_{2}$ .

~15 minProof · checked by AIcontinued fractionsFibonacci numbers

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XXVII. Recurring Continued Fractions, Examples XXVII.b, No. 16

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