國小數學題~乘方題~求解

[匿名]

2011-10-19 10:36 pm

有一群細胞，每小時死亡兩個

剩下的細胞每個分裂為兩個

設最初有7個細胞

則100個小時後有幾個細胞?

(請用乘方表示)

回答 (2)

善琁先生

2011-10-20 5:31 am

✔ 最佳答案

下午剛解過 :

f(1) = (7-2)*2 = 7*2 - 2^2
f(2) = ( f(1) -2 ) *2 = ( 7*2 - 2^2 -2)*2 = 7*2^2 - 2^3 - 2^2
= 7*2^2 - (2^3 + 2^2)
f(3) = ( f(2) -2 ) *2 = 7*2^3 - (2^4 + 2^3 + 2^2)
...f(100) = ( f(99) -2) *2 = 7*2^100 - (2^101 + 2^100 + 2^99 + ....+2^2)
2^101 + 2^100 + 2^99 + ....+2^2
= 2* (1+ 2 + 2^2 + 2^3 + ... 2^100)
= 2* [2* (2^100 -1)/ (2-1) ] [等差級數公式]
= 4*2^100 - 4
f(100) = 7*2^100 - (2^101 + 2^100 + 2^99 + ....+2^2)
= 7*2^100 - (4*2^100 - 4)
= 3*2^100 + 4 (Ans)註 :

若不帶等差級數公式, 也可以這樣算
S = 2^101 + 2^100 + 2^99 + ....+2^2
2*S = 2^102 + 2^101 + 2^100 + ......+2^3
= 2^102 + S - 2^2
S = 4*2^100 - 4

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老怪物

2011-10-20 2:40 am

這是 "國小" 數學題?

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