Re: [中學] 幾題數列級數和指對數請教

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※ 引述《izzylf (風)》之銘言：
: 各位版友好
: 這裡有幾題數列級數和指對數問題請教
: 因為符號不好在bbs顯現,所以用printscreen的
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: http://www.wretch.cc/album/show.php?i=izzylf0612&b=3&f=1619378913&p=1


1.   S  = 1 + (1+2) + (1+2+2^2) + ... + [1+2+2^2+...+2^(n-1)]

          n-1             n-1
        = Σ[2^(k+1)-1] = Σ2^(k+1) - n = 2^(n+1) - 2 - n
          k=0             k=0

2. S_n = 1/1 + 1/(1+2) + ... + 1/(1+2+...+n)

         n               n
       = Σ2/[(k+1)k] = 2Σ[1/k - 1/(k+1)] = 2 - 2/(n+1) → 2 as n→∞
         k=1             k=1

   |S-S_n|<0.01 => 2/(n+1)<0.01 => n > 2/0.01 - 1 = 199

   n最小值=200

3. a = log_60(3), b = log_60(5)

   [1-a-b]/[2-2b] = [1-log_60(3)-log_60(5)]/[2-2log_60(5)]

                  = log_60[60/(3x5)]/log_60[3600/25]

                  = log_60(4)/log_60(144)

                  = log_12(4)/log_12(144) = log_12(2)

   Thus, 12^[(1-a-b)/(2-2b)] = 2


4. a^2 - 1 = (1/4)[47^(2/2010)+47^(-2/2010)]+1/2-1
           = {(1/2)[47^(1/2010)-47^(-1/2010)]}^2

   Then, Ans = [47^(-1/2010)]^2010 = 1/47

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                                            ～by Jackary P.～

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