MathDB
Factorial Base Expansions

Source: AIME 2 2000 #14

December 9, 2005
factorial

Problem Statement

Every positive integer kk has a unique factorial base expansion (f1,f2,f3,…,fm),(f_1,f_2,f_3,\ldots,f_m), meaning that k=1!⋅f1+2!⋅f2+3!⋅f3+⋯+m!⋅fm, k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m, where each fif_i is an integer, 0≤fi≤i,0\le f_i\le i, and 0<fm.0<f_m. Given that (f1,f2,f3,…,fj)(f_1,f_2,f_3,\ldots,f_j) is the factorial base expansion of 16!−32!+48!−64!+⋯+1968!−1984!+2000!,16!-32!+48!-64!+\cdots+1968!-1984!+2000!, find the value of f1−f2+f3−f4+⋯+(−1)j+1fj.f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j.
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