TMUA 2017 Paper 2 Q15 — Worked Solution

P2·Number Theory & Divisibility, Sequences & Series·6.5Official · 2017 · Q15

collatz-likeperiodic-orbitmodular-position

The function

f

is defined on the positive integers as follows:

f (1) = 5

, and for

n \geq 1

f (n + 1) = 3 f (n) + 1

f (n)

is odd,

f (n + 1) = \frac{1}{2} f (n)

f (n)

is even.The function

g

is defined on the positive integers as follows:

g (1) = 3

, and for

n \geq 1

g (n + 1) = g (n) + 5

g (n)

is odd,

g (n + 1) = \frac{1}{2} g (n)

g (n)

is even.What is the value of

f (1000) - g (1000)

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