Fibonacci and squares

Algebra Level 4

The Fibonacci sequence $(F_n)$ is defined as: $F_1 = 1,\ F_2 = 1,\ F_n = F_{n-1} + F_{n-2}\ \text{for}\ n \geq 3.$

Find the sum of all positive integers $n$ such that $F_n=n^2$ .

1 solution

Khang Nguyen Thanh
Jan 22, 2017

We will prove that $F_n>n^2$ for all $n\ge13$ by induction $(*)$ .

For $n=13$ and $n=14$ , $(*)$ is true, since $F_{13}=233,F_{14}=377$ .

Suppose $(*)$ is true for some $n=k$ and $n=k+1$ with $k \ge 13$ .

We need prove that $(*)$ is true for $n = k + 2$ .

We have: $F_{k+2}=F_{k+1}+F_k>(k+1)^2+k^2=2k^2+2k+1=(k+2)^2+(k-3)(k+1)>(k+2)^2$ .

So, $F_n>n^2$ for all $n\ge13$ .

Consider $n\le 12$ , we get only 2 solutions, which are $F_1=1$ and $F_{12}=144$ .

Hence, the answer is: $1+12=\boxed{13}$ .

this is completely hit and trial at 2nd last step

Md Zuhair - 4 years, 4 months ago

We dont get any general equation, by which we get the value, But yes, It is a good sum

Md Zuhair - 4 years, 4 months ago