TMUA Official 2017 Paper 2 · Question 15

The function $f$ is defined on the positive integers as follows:

f(1)=5,\text{ and for }n\geq1:\qquad \begin{aligned} f(n+1)&=3f(n)+1&&\text{if }f(n)\text{ is odd}\\ f(n+1)&=\tfrac12f(n)&&\text{if }f(n)\text{ is even} \end{aligned}

The function $g$ is defined on the positive integers as follows:

g(1)=3,\text{ and for }n\geq1:\qquad \begin{aligned} g(n+1)&=g(n)+5&&\text{if }g(n)\text{ is odd}\\ g(n+1)&=\tfrac12g(n)&&\text{if }g(n)\text{ is even} \end{aligned}

What is the value of $f(1000)-g(1000)$ ?

-6

-5

1

2

4

8

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