# 2015 Countdown Problem #7: Cardinal Number Lengths

**Algebra**Level 3

*0<N≤2015*, let

*f(N)*denote the number of letters when N is spelt in English.

For example, 77 is spelt “*seventy seven*”, hence \[f(77)=7+5=12\]and \[f^2 (77)=f(12)=6\]
Also, 2015 is spelt “*two thousand and fifteen*”, hence \[f(2015)=3+8+3+7=21\].

Determine the value of \[\sum_{N=1794}^{2015} (\lim _{ k\rightarrow \infty }{ f^ { k} (N) } )\]

*This problem is part of the set 2015 Countdown Problems.*