自我練習 2:
如圖,A柱中有n個大小不同的圓盤由大而小往上堆
疊,若要從A柱全部搬移至B柱,每次只能搬動一圓盤,
且每次都必須先經中間柱(不可由A直接放入B)且大盤
不可放在小盤之上,設共要搬動 an次,若 an + 1 = pan + k,
求數對(p,k) = (36) *公式9n=27-1
(>
591 3112 =) #6 an= 2³² -1 = 1) =R
次
an=2²-1 = 3 =2
◆◆◆解答
1. 解
(1) a = 1,
a2 = 2a₁ + 1 = 3 = 2² - 1,
a3 = 2a2 + 1 = 7 = 2³ – 1,
a = 2a3 + 1 = 15 = 2 −
-
1,
as = 2a + 1 = 31 = 2 - 1,
推測一般項 an = 2" - 1(n 為正整數),
(2)由數學歸納法證明 an = 2" - 1
:
24=16-1
①當n=1 時, a = 2' - 1 = 1,原式成立.
ok
1個 = 2'-1 = 1
äi- | √axler
Az = 3
↓3x2+13=7.22*an-1-
die 1, the
92= 3²² 1×2+1
93= 2
94=15
C
B
h=2²
an= zan-1 +1
Gn+1 = 2an+ |
