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If f :N→N such that f(f(x))=3x;
Then find f(2013)

10 Answers

2305
Shaswata Roy ·

Is it 3852?

15
Anomitro Kundu ·

3852

2305
Shaswata Roy ·

Sorry I didn't post the full solution before.I had taken an assumption which I cannot prove.

The assumption was that f(x) is a strictly increasing function.

Anyways assuming that is true,here are some of the values of f(x).

f(n)≠n for all n.Because that would imply f(f(n))=n but we know that f(f(n))=3n.

Furthermore f(n)>n.
Because if f(n)<n then f2(n)<f(n) (since f(x) is increasing)
→3n<f(n).This leads to a contradiction.

From the above we can see that,

f(n)>n and f2(n)=3n>f(n)

→ 3n>f(n)>n

For n=1,

3>f(1)>1 →f(1)= 2

f(2) = f2(1) = 3

f(3) = f2(2) = 6

f(6) = f2(3) = 9

f(3)<f(4)<f(5)<f(6)
→6<f(4)<f(5)<9

Therefore,f(4) = 7 and f(5) = 8

After doing this for the first 20 numbers I found a pattern.

From here we can see that if

3k<n<2*3k then,

f(n) = n+3k

and if 2*3k<n<3k+1 then,

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

You can prove this using induction.

Hence f(2013) = 3852

However the solution is not complete.Someone still needs to prove that f(n) is strictly increase(proving that it is strictly monotonic also can be helpful) because I'm pretty sure that the function is strictly increasing.

466
Himanshu Giria ·

no one for this qn...??

466
Himanshu Giria ·

No 1 for this question???

466
Himanshu Giria ·

No 1 for this question???

466
Himanshu Giria ·

No 1 for this question???

486
Niraj kumar Jha ·

f(x)=x√3 (Just observed it. I don't know the method)
f(2013)=2013√3

591
Akshay Ginodia ·

From where did u get this qstn?

466
Himanshu Giria ·

thanks

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