中四A-MATH題@@

2008-11-12 6:30 am

Let a1 = 1 and an+1 = an+n for n>=1.Prove that an = 1/2n(n-1)=1 for all positive integers n.

(連住a的數全部放在a下面,不是乘入去=0=)

有人識計嗎=.=
最好詳細一點@@
謝謝

回答 (1)

Uncle Michael

2008-11-12 7:09 am

✔ 最佳答案

P(n): an = (1/2)n(n-1)+1 for all positive integers n

where a1 = 1 and an+1 = an + n

When n = 1:

a1 = 1

(1/2)(1)(1-1) + 1 = 1

Hence, P(n) is true.

Assume that n = k is true, i.e.

ak = (1/2)k(k-1)+1

When n = k+1:

ak+1

= ak + (k+1)

= (1/2)k(k-1)+1+(k+1)

= [(1/2)k(k-1)+(k+1)]+1

= (1/2)[k(k-1)+2(k+1)]+1

= (1/2)[k2-k+2k+2]+1

= (1/2)(k2+k+2)+1

= (1/2)(k+1)(k+2)+1

= (1/2)(k+1)[(k+1)+1]+1

Hence, P(k+1) is true when assuming P(k) is true.

According to the principle of mathematical induction,

P(n) is true for all positive integers n

where a1 = 1 and an+1 = an + n
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