HMMT 十一月 2008 · 冲刺赛 · 第 18 题

HMMT November 2008 — Guts Round — Problem 18

[ 10 ] Find the coefficient of x in the expansion of 6 ∑ 6 i ( x + 1) · x i =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . st 1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND

解析

[ 10 ] Find the coefficient of x in the expansion of 6 ∑ 6 i ( x + 1) · x i =0 3 6 i Answer: 64 Each term of ( x + 1) can be multiplied by a unique power x , 0 ≤ i ≤ 6 to get a sixth 6 degree term. So the answer is the sum of the coefficients of the terms of ( x + 1) , which is the same as 6 substituting x = 1 into this power to get 2 = 64. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . st 1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUND