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 \(F_{100}\) odd or even?

Is \(F_{100}\) a multiple of \(3\)?

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

Among the first \(20\) Fibonacci numbers: \(F_0 = 0,F_1 = 1,F_2 = 1, F_3 = 2, F_4 = 3,..., F_{20} = 6765\) find all the numbers, which have sum of digits equal to their index. For example \(F_1=1\) fits the description, however \(F_{20} = 6765\) does not, since \(6+7+6+5 \neq 20\).

Among the first \(20\) Fibonacci numbers: \(F_0 = 0,F_1 = 1,F_2 = 1, F_3 = 2, F_4 = 3,..., F_{20} = 6765\) check whether the numbers with prime index are prime.

Consider the 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 the Pascal’s triangle sum up to Fibonacci numbers.

Prove for any \(m,n\) that \(F_{m+n} = F_{m-1}F_n + F_mF_{n+1}\).

Denote by \(GCD(m,n)\) the greatest common divisor of numbers \(m,n\), namely the largest possible \(d\) which divides both \(n\) and \(m\). Prove for any \(m,n\) that \[GCD(F_n,F_m) = F_{GCD(m,n)}.\]