Summation of Fibonacci number multiplied by a variable

Question

I want to find a general formula for calculating:

$$\sum_{i=1}^n i F_i $$

where $F_n$ is the n-th Fibonacci number. What I have done so far is: $$\sum_{i=1}^n \sum_{j=i+1}^n F_j$$

I tried to solve it using above double summation formula but it doesn't give the correct result

$F_1$ = 1, $F_2$ = 2, $F_3$ = 3, $F_4$ = 5.

(for e.g. summation upto 12 is 3916, upto 5 is 46)

Answer 1

If you want to compute the sum $$\sum_{i=1}^n iF_i$$ you can write $$\sum_{i=1}^n iF_i =\sum_{i=1}^n \sum_{j=1}^i F_i = \sum_{j=1}^n \sum_{i=j}^n F_i $$ and switching $i,j$ it leads to $$ \sum_{i=1}^n \sum_{j=i}^n F_j$$ If you want to achieve a close form, you may remember $$F_1+F_2+F_3+\dots+F_k = F_{k+2}-2$$ so $$ \sum_{i=1}^n \sum_{j=i}^n F_j = \sum_{i=1}^n (F_{n+2}-2 -\sum_{j=1}^{i-1} F_j ) = nF_{n+2} - 2n - \sum_{i=1}^n(F_{i+1}-2) $$ $$= nF_{n+2} - F_{n+3}+2+F_1 =nF_{n+2} - F_{n+3} + 3 $$

Answer 2

Using finite differences:

Let $f(n) = \sum_{i=1}^{n} iF_i$

$\Delta f(n) = (n+1)F_{n+1} = \Delta (nF_{n+2} - F_{n+3})$

$\implies f(n) = nF_{n+2} - F_{n+3} + c$

For $n=1$, $f(1) = F_1 = 1 = F_3 - F_4 + c \implies c = 3$

Hence $f(n) = nF_{n+2} - F_{n+3} + 3 $