I have a sequence with explicit formula $b_n = F_{n+4} -(n+3)$ The question asks me to use that formula to hence find the following sum for each $n$: $$nF_1 + (n-1)F_{2} +(n-2)F_3 +...+2F_{n-1}+F_n $$ I feel like the answer to my question will be very obvious, but I really don't know where to start with it.
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Is the sum equal to $b_n$? If so, you can probably prove it using induction. – Christopher Carl Heckman Apr 25 '16 at 03:28
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I have that $b_n = F_{n+4} - (n+3)$ which can be changed to $b_n = 2F_n +3F_{n+1} - (n+3)$. But I don't think I can get that strictly in terms of $b_n$. – Nico Monk Apr 25 '16 at 04:06
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@NicoMonk: Induction works to prove that $nF_1+(n-1)F_2+\cdots +F_n=F_{n+4}-(n+3)\ (=b_n)$. Don't you want to use induction? – mathlove Apr 25 '16 at 06:45
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@mathlove I wasn't sure if induction would work here. I am as happy to use induction as the next guy on this website. I just wasn't sure if $b_n$ could be made to equal the sum. Like I said above, the solution was probably very obvious. – Nico Monk Apr 25 '16 at 10:30