Big Fibonacci
The fifth term of a Fibonacci sequence is 2004. If all the terms are positive integers, what is the largest possible first term?
Problem
In a sequence of positive integers, every term after the first two terms is the sum of the two previous terms in the sequence.
If the fifth term is $2004$, what is the maximum possible value of the first term?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $999$
One unknown
The first term will be biggest when the second term is smallest. Try:
$\underline{\ \ \ n\ \ \ },\quad\underline{\ \ \ 1\ \ \ }, \quad \underline{\qquad\ }, \quad \underline{\qquad\ }, \quad \underline{2004}$
$\underline{\ \ \ n\ \ \ },\quad\underline{\ \ \ 1\ \ \ }, \quad \underline{n+1}, \quad \underline{n+2}, \quad \underbrace{\underline{2004}}_{2n+3}$
$2n+3=2004 \Rightarrow 2n=2001$ which is odd
Try:
$\underline{\ \ \ n\ \ \ },\quad\underline{\ \ \ 2\ \ \ }, \quad \underline{\qquad\ }, \quad \underline{\qquad\ }, \quad \underline{2004}$
$\underline{\ \ \ n\ \ \ },\quad\underline{\ \ \ 2\ \ \ }, \quad \underline{n+2}, \quad \underline{n+4}, \quad \underbrace{\underline{2004}}_{2n+6}$
$2n+6=2004 \Rightarrow 2n=1998\Rightarrow n=999$
Two unknowns
$\underline{\ \ \ \ a\ \ \ \ },\quad\underline{\ \ \ \ b\ \ \ \ }, \quad \underline{\ \ \qquad\ }, \quad \underline{\qquad\ \ \ }, \quad \underline{\ 2004\ }$
$\underline{\ \ \ \ a\ \ \ \ },\quad\underline{\ \ \ \ b\ \ \ \ }, \quad \underline{\ a+b\ }, \quad \underline{a+2b}, \quad \underbrace{\underline{\ 2004\ }}_{2a+3b}$
So $2a+3b = 2004$
$a$ large if $b$ small
If $b=1$ then $2a=2001$, but $a$ is an integer, so $b\not=1$.
However, if $b=2$ then $2a=1998$, so the maximum possible value of $a$ is $999$