Math CentralQuandaries & Queries


Question from nahla, a student:

Hello !! I am not a native english speaker and I am currently at grade 11. I have done all my maths studies in arabic, but there has been this problem that I came across and that really bothered me, I cannot seem to find the answer, although I have some hints of how it should be done, I cannot seem to get it right, here it is, if you do not mind clearing it up !

f: IN --> IN
n --> f(n)
for every n that belongs the IN : fof(n) = 4n - 3
and for every n that belongs to IN f(2^n) = 2^(n+1) -1

Calculate f(993)

Thank you so much ! And greetings !

Hi Nahla,

I'll give you some hints also and hopefully they will help.

Start with the first condition and see if $993$ can be the result of $f \circ f(n)$ because if is is than $f(993) = f \circ f \circ f(n).$

If $f \circ (n) = 993$ then $4n - 3 = 993$ and $n = 249.$ Hence $f \circ f(249) = 993.$

What about $249?$

If $f \circ (n) = 249$ then $4n - 3 = 249$ and $n = 63.$ Hence $f \circ f(63) = 249.$

But $63$ is one less than a power of $2$ and that makes me think of the second condition, $f(2^n) = 2^{n+1} -1.$

A fact that is useful is that $f \circ f \circ f(n) = f \circ f \left(f(n) \right) = f \left( f \circ f(n) \right).$


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