S6 Maths. Sequences and Series

(a) Let f(x) = (x+1)^[n(2n+1)], so the remainder when f(x) is divided by x is :
f(0)
= (0+1)^[n(2n+1)]
= 1

(b) Pn
= 8 * 8^2 * 8^3 * ... * 8^2n
= 8^(1+2+3+...+2n)
= 8^[(1+2n)(2n)/2]
= 8^[n(2n+1)]
(c) A week has 7 days, to calc the day of the week is to find the remainder of Pn divided by 7. Put x=7 into part (a), therefore Pn=f(x)=f(7), so the remainder is 1. 1 day after Monday is Tuesday. That is, the day of the week after Pn days is Tuesday.

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吴匑哀仆俪

8ⁿ = (1 + 7)ⁿ = 1 + 7K

a)(x+1) ^ n(2n+1) / x = (x+1) ^ [n(2n+1)-1] remainder = 0

b) Pn=8 x 8^2 x 8^3 x ... x 8^2n = 8^(1+2+3+ ....... + 2n)
= 8^[(1+2n)(2n)/2]

c) 唔識做，計日子應該是 +7+7+7之類，例如 1 號是星期一，則 8 號，15號，22號也是星期一，唔知點同 8 拉上關係。

2013-11-30 00:12:34 補充：
用 Binomial 將式子改為 (1+7)^m 的形式，可得到

(1+7)m = 1 + (m)(7) + [(m)(m-1)/2](7^2) + [(m)(m-1)(m-2)/(3)(2)](7^3) + ..................

所以答案是 星期二(Tuesday)。