# 2015 Countdown Problem #7: Cardinal Number Lengths

Algebra Level 4

For every integer 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.

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