## Task 3 - 340523

Every natural number $1, 2, 3, \ldots$ can be represented as a sum where each summand is either 1 or 2. For example, for the number 3, considering the order, there are exactly the representations

$$
3=1+1+1=1+2=2+1
$$

(a) Give all representations for each of the numbers 4, 5, and 6 as well!

(b) How many representations are there for each of the numbers 1, 2, 3, 4, 5, and 6, respectively?

Find a pattern that holds for these six counts!

How many representations must there be for the number 10 if the pattern you mentioned holds generally?
(a) The sought representations are:

$$
\begin{gathered}
4=1+1+1+1=1+1+2=1+2+1=2+1+1=2+2 \\
5=1+1+1+1+1=1+1+1+2=1+1+2+1=1+2+1+1=2+1+1+1=1+2+2= \\
=2+1+2=2+2+1 \\
6=1+1+1+1+1+1=1+1+1+1+2=1+1+1+2+1=1+1+2+1+1= \\
=1+2+1+1+1=2+1+1+1+1=1+1+2+2=1+2+1+2=1+2+2+1= \\
=2+1+1+2=2+1+2+1=2+2+1+1=2+2+2
\end{gathered}
$$

(b) The counts are:

$$
\begin{array}{ccccccc}
\text { Number to be represented } & 1 & 2 & 3 & 4 & 5 & 6 \\
\text { Number of representations } & 1 & 2 & 3 & 5 & 8 & 13
\end{array}
$$

The following pattern applies to these counts:

The sum of two consecutive counts is equal to the next count. If this pattern holds generally, the continuation is:

$$
\begin{array}{ccccc}
\text { Number to be represented } & 7 & 8 & 9 & 10 \\
\text { Number of representations } & 8+13=21 & 13+21=34 & 21+34=55 & 34+55=89
\end{array}
$$

For the number 10, there must then be exactly 89 representations.