How to solve this recurrence relation f(x,0)=3x and f(x,n+1)=f(f(x,n),n)?

Humberto Campbell

Humberto Campbell

Answered question

2022-11-11

How to solve this recurrence relation f ( x , 0 ) = 3 x  and  f ( x , n + 1 ) = f ( f ( x , n ) , n ) and f ( x , 0 ) = 3 x  and  f ( x , n + 1 ) = f ( f ( x , n ) , n )?

Answer & Explanation

mainzollbtt

mainzollbtt

Beginner2022-11-12Added 13 answers

Take x = 2005 and n = 2, then f ( 2005 , 3 ) = f ( f ( 2005 , 2 ) , 2 )
Take x = 2005 and n = 1, then f ( 2005 , 2 ) = f ( f ( 2005 , 1 ) , 1 )
Take x = 2005 and n = 0, then f ( 2005 , 1 ) = f ( f ( 2005 , 0 ) , 0 )
Now f ( 2005 , 0 ) = 3 × 2005 = 6015
Thus f ( 2005 , 1 ) = f ( f ( 2005 , 0 ) , 0 ) = f ( 6015 , 0 ) = 3 × 6015 = 18045
f ( 2005 , 2 ) = f ( f ( 2005 , 1 ) , 1 ) = f ( 18045 , 0 ) = 3 × 18045 = 54135
f ( 2005 , 3 ) = f ( f ( 2005 , 2 ) , 2 ) = f ( 54135 , 0 ) = 3 × 54135 = 162405
In general:
f ( x , 0 ) = 3 x , f ( x , 1 ) = 3 2 x , . . . , f ( x , n ) = 3 n + 1 x
f ( 2005 , 3 ) = 3 4 2005 = 162405

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