1. Let [tex]s_n[/tex] be the number of seats in the [tex]n[/tex]-th row. The number seats in the [tex]n[/tex]-th row relative to the number of seats in the [tex](n-1)[/tex]-th row is given by the recursive rule
[tex]s_n=s_{n-1}+5[/tex]
Since [tex]s_1=21[/tex], we have
[tex]s_2=s_1+5[/tex]
[tex]s_3=s_2+5=s_1+2\cdot5[/tex]
[tex]s_4=s_3+5=s_1+3\cdot5[/tex]
[tex]\cdots[/tex]
[tex]s_n=s_{n-1}+5=\cdots=s_1+(n-1)\cdot5[/tex]
So the explicit rule for the sequence [tex]s_n[/tex] is
[tex]s_n=21+5(n-1)\implies s_n=5n+16[/tex]
In the 15th row, the number of seats is
[tex]s_{15}=5(15)+16=91[/tex]
2. Let [tex]p_n[/tex] be the amount of profit in the [tex]n[/tex]-th year. If the profits increase by 6% each year, we would have
[tex]p_2=p_1+0.06p_1=1.06p_1[/tex]
[tex]p_3=1.06p_2=1.06^2p_1[/tex]
[tex]p_4=1.06p_3=1.06^3p_1[/tex]
[tex]\cdots[/tex]
[tex]p_n=1.06p_{n-1}=\cdots=1.06^{n-1}p_1[/tex]
with [tex]p_1=40,000[/tex].
The second part of the question is somewhat vague - are we supposed to find the profits in the 20th year alone? the total profits in the first 20 years? I'll assume the first case, in which we would have a profit of
[tex]p_{20}=1.06^{19}\cdot40,000\approx121,024[/tex]
3. Now let [tex]p_n[/tex] denote the number of pushups done in the [tex]n[/tex]-th week. Since [tex]3\cdot4=12[/tex], [tex]12\cdot4=48[/tex], and [tex]48\cdot4=192[/tex], it looks like we can expect the number of pushups to quadruple per week. So,
[tex]p_n=4p_{n-1}[/tex]
starting with [tex]p_1=3[/tex].
We can apply the same reason as in (2) to find the explicit rule for the sequence, which you'd find to be
[tex]p_n=4^{n-1}p_1\implies p_n=4^{n-1}\cdot3[/tex]
