Question

Suppose that $f\left(n\right)$ is a real valued function whose domain is the set of positive integers and that $f\left(n\right)$ satisfies the following two properties:
$f\left(1\right)=23$ and $f(n+1)=8+3.f\left(n\right)$, for $n\ge 1$
It follows that there are constant $p,q$ and $r$ such that $f\left(n\right)=p.q^n-r$, for $n=1,2,......$ then the value of $p+q+r$ is:

• A. $16$
• B. $17$
• C. $20$
• D. $26$
• E. $31$

Answer 1

Answer: (A)

$16$

$f\left(n\right)=p{q}^n-r$
$f\left(1\right)=23=pq-r\Rightarrow r=pq-23$ ...(1)
$f\left(2\right)=f(1+1)=8+3.f\left(1\right)=8+3.23=77=p{q}^2-r\Rightarrow r=p{q}^2-77$ ...(2)
$f\left(3\right)=f(2+1)=8+3.f\left(2\right)=8+3.77=239=p{q}^3-r\Rightarrow r=p{q}^3-239$ ...(3)
Solving (1) and (2)
$pq-23=p{q}^2-77\Rightarrow pq(q-1)=54$ ...(4)
Solving (2) and (3)
$p{q}^2-77=p{q}^3-239\Rightarrow p{q}^2(q-1)=162$ ...(5)
Now from (4) and (5)
$q=3,p=9,r=4\Rightarrow p+q+r=16$