论文部分内容阅读
1982年高考数学试卷(理科)第九题是: 已知数列a_1,a_2,…,a_n,…和数列b_1,b_2,…,b_n,…,其中a_1=p,b_1=q,a_n=pa_(n-1),b_n=qa_(n-1)+rb_(n-1)(n≥2),(p,q,r是已知常数,且q≠0,p>r>0)。 (1)用p,q,r,n表示b_n,并用数学归纳法加以证明; (2)求limb_n/(a~2+b~2)~(1/2) 。该题(1)解题过程有以下几个步骤: 1.尝试:∵a_1=p,a_n=pa_(n-1),∴a_n=p~n,b_1=q,b_2=qa_1+rb_1=q(p+r),b_3=qa_2+rb_2=q(p~2+pr+r~2)…… 2.观察:b_1、b_2、b_3的表达式都是q和p、r齐次式的乘积。 3.猜想:b_n=q(p~(n-1)+p~(n-1)1+……+r~(n-1))。 4.论证:(用数学归纳法)从略。这是一个完整的逻辑推理过程,前一半是用简
The ninth question of 1982 college entrance examination math test (science) is that the known series a_1, a_2, ..., a_n, ... and the sequence b_1, b_2, ..., b_n, ... where a_1 = p, b_1 = q and a_n = pa_ (p, q, r are known constants, and q ≠ 0, p> r> 0), b_n = qa_ (n-1) + rb_ (n-1) (2) Find limb_n / (a ~ 2 + b ~ 2) ~ (1/2) by using p, q, r, n for b_n and prove it by mathematical induction. The problem (1) solving process has the following steps: 1. Try: ∵a_1 = p, a_n = pa_ (n-1), ∴a_n = p ~ n, b_1 = q, b_2 = qa_1 + rb_1 = q (p + r), b_3 = qa_2 + rb_2 = q (p ~ 2 + pr + r ~ 2) ...... 2. Observe that the expressions of b_1, b_2 and b_3 are both q and p, . 3. Conjecture: b_n = q (p ~ (n-1) + p ~ (n-1) 1 + ... + r ~ (n-1)). 4. Argument: (using mathematical induction) omitted. This is a complete logical reasoning process, the first half is using Jane