a-maths~~~~~~~~

(a)

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(b) (i)

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I don't know how to answer in Chinese.
a) Let P(n) be the proposition "13 + 23 +33 + ... + n3 = n2 (n+1)2 / 4".
When n = 1, L.H.S. = 13
R.H.S. = 1 (1+1)2 /4 = 1 = L.H.S.
So P(1) is true.
Assume that P(k) is true, i.e. "13 + 23 +33 + ... + k3 = k2 (k+1)2 / 4"
When n = k + 1, L.H.S.
= 13 + 23 +33 + ... + k3 + (k + 1)3
= k2 (k+1)2 /4 + (k + 1)3
= (1/4)(k + 1)2 (k2 + 4k + 4)
= (k + 1)2 (k + 2)2 / 4
= R. H. S.
So P(k + 1) is true if P(k) is true.
By Mathematical Induction, P(n) is true for all positive integers n.
bi) No. of bricks in the rth "floor" = r2
ii) Taking away 1 brick in rth "floor" needs r minutes. Then taking away all the bricks in the rth "floor" needs r x r2 minutes, i.e. r3 minutes.
So, time needed to take away all the bricks from 1th to 10th "floor"
= 13 + 23 + 33 + ... + 103
= 102(10 + 1)2/4
= 3025 minutes
Time needed to take away bricks in the next 10 "floor"
= 13 + 23 + 33 + ... + 203 - (13 + 23 + 33 + ... + 103 )
= 202(20 + 1)2 / 4 - 3025
= 41075 minutes