Problem
ALG-B3-M02-P005 Recurrence from functional form
#5
★★★☆☆ Level 3 of 5
Let \(f:\mathbb N\to\mathbb N\), \(f(n+1)=f(n)+2\), and \(f(f(1))=5\). Find \(f(n)\).
Hint. Let \(f(1)=c\), then \(f(n)=c+2(n-1)\).
The recurrence gives \(f(n)=c+2(n-1)\). Then \(f(f(1))=f(c)=c+2(c-1)=3c-2\). The condition gives \(3c-2=5\), so \(c=7/3\), not a positive integer. Therefore no such function exists.
Goal: choose the right first substitution and then verify the found function.