Problems

What’s the sum of the Fibonacci numbers \(F_0+F_1+F_2+...+F_n\)?

What’s the sum \(\frac{F_2}{F_1}+\frac{F_4}{F_2}+\frac{F_6}{F_3}+...+\frac{F_{18}}{F_9}+\frac{F_{20}}{F_{10}}\)?

Is \(F_{23}\) odd or even?

Is \(F_{24}\) a multiple of \(3\)?

Let \(n\) be a positive integer. Can \(n^7-77\) ever be a Fibonacci number?

Consider Pascal’s triangle: it starts with \(1\), then each entry in the triangle is the sum of the two numbers above it. Prove that the diagonals of Pascal’s triangle add up to Fibonacci numbers.