$15$ points

$(5)$ Due to births, a lake fish population increases yearly by $13\%$; mortality decreases the population by $3\%$ yearly. Each year, fishing is allowed until $1,000$ fish are caught. Thereafter, fishing is banned. Currently there are $12,000$ fish in the lake.

$($a$)$ Write the difference equation relating $x_{n+1}$ to $x_n$ for the lake fish population.

$-7.$

$($b$)$ Find the general solution to the difference equation. $x_0=12,000$

$10$ points

$(6)$ Find the value of $x_8$ given $x_{n+1}=x_n+12$ and $x_0=150.$

$x_1=x_0+12=150+12=162$

$x_2=x_1+12=162+12=174$

$x_3=x_2+12=174+12=186$

$x_4=x_3+12=186+12=198$

$x_5=x_4+12=198+12=20$

$x_6=x_5+12=210+12=222$

$x_7=x_6+12=222+12=234$

$x_8=x_7+12=234+12=246$

with $x_0=150$

$x_6=246$

WHSN$\text{'}$T

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