## SUBJECT I

a). At a match, a number of people enter the stadium according to the following rule: in the first minute after the stadium opens, one person enters, in the second minute, twice the number of people from the first minute plus one more person enter, and the rule continues, meaning in minute $\mathrm{k}+1$, twice the number of people from minute $\mathrm{k}$ plus one more person enter, $\mathrm{k} \geq 1$. How many people have entered after seven minutes?

b). The sequence $1,3,7,15,31,63, \ldots$ is given. Show that if $n$ is the 201st term of the sequence, then the number $n+1$ is a perfect cube.
## SUBJECT I

a) In the first minute, one person

In the second minute $2 \cdot 1+1=3$ people

In the third minute $2 \cdot 3+1=7$ people

In the fourth minute $2 \cdot 7+1=15$ people

In the fifth minute $2 \cdot 15+1=31$ people

In the sixth minute $2 \cdot 31+1=63$ people

In the seventh minute $2 \cdot 63+1=127$ people 2 points

After seven minutes, $1+3+7+15+31+63+127=247$ people enter 1 point
b) $t_{1}=1$

$t_{2}=2 \cdot 1+1=3$

$t_{3}=2 \cdot(2+1)+1=2^{2}+2+1$

$t_{4}=2 \cdot\left(2^{2}+2+1\right)+1=2^{3}+2^{2}+2+1$

1 point

$\left.\mathrm{t}_{201}=2^{200}+2^{199}+\cdots+2+1=\mathrm{n} \quad\right\}$

2 points

Thus $n=2^{201}-1 \Rightarrow n+1=2^{201}$

$\mathrm{n}+1=\left(2^{3}\right)^{67}=\left(2^{67}\right)^{3}=$ perfect cube $\} \quad 1$ point