a) Prove, by mathematical induction, that 1^2+2^2+...+n^2=1/6(n)(n+1)(2n+1) for all positive integers n.
b)Using the formula in (a), find the sum 1*2+2*3+...+n(n+1).
a題已prove, 唔識b題。
中四M2 Mathematical induction
2014-04-01 8:02 am
回答 (1)
2014-04-01 8:13 am
✔ 最佳答案
1*2 + 2*3 + ... + n(n+1)= 1*(1+1) + 2*(2+1) + ... + n(n+1)= 1*1 + 1 + 2*2 + 2 + ... + n*n + n= (1^2 + 2^2 + ... + n^2) + (1 + 2 + ... + n)= n(n+1)(2n+1)/6 + n(n+1)/2= [n(n+1)/6]*(2n+1 + 3)= n(n + 1)(n + 2)/3
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