Arithmetic Series, when $n$ tends to infinity the limit is $24$

Question

The $n$-th term of a sequence is $U_n$

$$U_{n+1}=pU_n+q$$

• $p$ and $q$ are constants
• the first two terms are $U_1=96$ and $U_2=72$
• the limit as $n$ tends to infinity is $24$

a) show that $p=2/3$

b) find the value of $U_3$

Answer 1

(1): $72=96*p+q$

$lim_{n\to\infty}U_n=lim_{n\to\infty}U_{n+1}$.

(2): $24=24*p+q$

Solving the two inequalities we get, $p=\frac{2}{3},q=8$

$U_3=\frac{2}{3}*72+8=56$