Fibonacci sequence

https://cdn.discordapp.com/attachments/322819040059850757/372793108338835463/unknown.png

I found the formula for the sum for this sequence and it turns out to be

$S_n =F_{n+2} -1$


So i tried to add the 2 numbers you see there and get 288 and take 1 you get 287 but of course this isn't the right number

so i decided to find the general solution which happened to be

$a_n =\dfrac{1}{\sqrt 5} \cdot \left(\dfrac{1+\sqrt 5}{2} \right)^n - \dfrac{1}{\sqrt 5} \cdot \left(\dfrac{1-\sqrt 5}{2} \right)^n$

and the 12th term happens to be 377 and i took 1 off of this and it's still wrong, i don't get where i'm going wrong.

$S_{12}=F_{14}-1$ so you need $n=14$ in the Fibonacci number formula.

Why do they say that $F_{12}=F_{13}=144$...? That's not right.

$F_{12}=144$ and $F_{13}=233$

Did you derive the correct relation for $S_n$?

It's possible there's some confusion between "nth term" and $a_n$, given they start the series with $a_0$.

That said, under their definitions we get the sum

1+1+2+3+5+8+13+21+34+55+89+144=376

I can't really see how that's debatable (i.e. if there is confusion, it's them who are confused...)

Your expression for $S_n$ is correct and your answer is definitely correct, if the value for $F_{13}$ was indeed a typo. I'm not sure how you could possible get $F_{12} = F_{13} = 144$ on the off-chance it's not a typo though...

Couple of notes: you're not "supposed" to just add it up, but note that it only really takes a few seconds to do this by hand. Worth remembering when you need to do this in an exam.

Similarly, the formula you give is probably more hassle than it's worth until about the 377 term.

Finally, for n > 2 (or so), you can ignore the 2nd term in the formula and instead just pick the closest integer to the 1st term. Saves time + effort.

HOLY MOLEY I WAS RIGHT? What is this, a miracle?

HAHA LOL YOU THOUGHT IT WAS A TYPO? LOOK AGAIN

(btw i had to refresh the page so different numbers came up)
https://cdn.discordapp.com/attachments/322819040059850757/372799528560230400/unknown.png


not quite sure i understand this

not quite sure i understand this

e.g.

n = 10, $\dfrac{1}{\sqrt{5}} \left(\dfrac{1+\sqrt{5}}{2} \right)^n$ = 55.0036361...; nearest integer is 55.

n = 20, $\dfrac{1}{\sqrt{5}} \left(\dfrac{1+\sqrt{5}}{2} \right)^n$ = 6,765.0000296...; nearest integer is 6765