Problems

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.

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?