Re: [中學] 兩題數學問題
※ 引述《waynan (waynan)》之銘言：
: 因為題目有多重根號，因此我用附圖的方式比較可以看清楚題目，謝謝!
: http://ppt.cc/oHl8
We denote r_k(a)=a^(1/2^k),
then Q1 is equal to estime the integer part of r_1(2)r_2(3)*...*r_2013(2014).
Since a+1 <= 2^(a-1) for a>=2, we have
r_a(a+1) <= 2^((a-1)/2^a) for all a>=2.
2013
Thus, r_2(3)r_3(4)...r_2013(2014) <= 2^(Σ(a-1)/2^a)
a=2
< 2.
Final, r_1(2)r_2(3)*...*r_2013(2014) < 2√2 < 3.
It's easy to see that the product > 2 by considering the first few factors.
Thus, [r_1(2)r_2(3)*...*r_2013(2014)]=2.
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