HMMT 十一月 2008 · 冲刺赛 · 第 9 题
HMMT November 2008 — Guts Round — Problem 9
题目详情
- [ 7 ] Let a = , and
{70
2 a if a <1
n na =2
n +1
2 a − 1 if a ≥ .1
n n
Find a .2
2008
1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUNDst 2
英文原题
[ 7 ] Let a 0 = 6
7 , and
a n +1 =
{
2 a n if a n < 1
2
2 a n − 1 if a n ≥ 1
2 .
Find a 2008 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 st HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND
解析
英文解析
- [ 7 ] Let a = , and
{70
2 a if a <1
n na =2
n +1
2 a − 1 if a ≥ .1
n n
Find a .2
2008
Answer: We calculate the first few a :5
7 i
5 3 6
a = , a = , a = = a
1 2 3 0
7 7 7
6 5
So this sequence repeats every three terms, so a = a = . Then a = .
2007 0 2008
7 7
1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUNDst 2