📚 Exercise 3H: Sequences and Series Practice for IB Math | 练习 3H：IB 数学数列与级数练习
In many IB Mathematics textbooks, Exercise 3H is a pivotal practice set within the chapter on sequences and series. This collection of problems challenges students to apply their understanding of arithmetic and geometric progressions, summation techniques, and real-world applications. Mastering these skills is essential for success in both Analysis & Approaches (AA) and Applications & Interpretation (AI) courses, as sequences appear in Paper 1 and Paper 2 with increasing complexity.
在许多 IB 数学教材中，练习 3H 是数列与级数章节中的关键练习集。这组题目要求学生运用等差数列、等比数列、求和方法以及实际应用等方面的知识。由于数列题目在试卷一和试卷二中频繁出现且难度递增，无论是分析数学 (AA) 还是应用数学 (AI) 课程，掌握这些技能对于取得成功至关重要。
1. Understanding the Scope of Exercise 3H | 了解练习 3H 的范围
Exercise 3H typically appears at the end of a chapter on sequences and series, serving as a mixed-review set. It blends basic skill drills, such as finding the 10th term of an arithmetic sequence, with applied questions like modelling depreciation or summing sigma notation expressions. The exercise often mirrors the style of IB exam questions, requiring clear working and justification of steps, especially when dealing with infinite geometric series and algebraic manipulation of sum formulas.
练习 3H 通常出现在数列与级数章节末尾，作为一种综合复习题组。它将基础技能训练（例如求等差数列的第 10 项）与应用题（如折旧建模或求和符号表达式求和）结合在一起。这些练习往往模仿 IB 考试题的风格，要求解题步骤清晰、有据可依，特别是在处理无穷等比级数和求和公式的代数操作时更为突出。
2. Arithmetic Sequences: Finding the nth Term | 等差数列：求第 n 项
An arithmetic sequence is defined by a constant difference d between consecutive terms. The general term is uₙ = u₁ + (n − 1)d , so d = 3, then u₁ = 1. This approach is essential for building models from partial data.
等差数列由相邻项之间的常数公差 d 所定义。其通项公式为 uₙ = u₁ + (n − 1)d。在练习 3H 中，你可能会遇到给出两个非连续项的情况。例如，若 u₃ = 7 且 u₇ = 19，我们可建立方程组：u₁ + 2d = 7 和 u₁ + 6d = 19。相减得 4d = 12，因此 d = 3，进而 u₁ = 1。这种方法是根据部分数据建立模型的关键。
3. Arithmetic Series: Summation Formulas | 等差数列求和公式
The sum of the first n terms of an arithmetic sequence is given by two equivalent forms: Sₙ = n⁄2 (u₁ + uₙ) or Sₙ = n⁄2 the number of terms when the sum is known, the second form becomes a quadratic in n: Sₙ = n⁄2 . Rearranging and solving via factorization or the quadratic formula yields n. Always verify that n is a positive integer.
等差数列前 n 项和可由两种等价形式给出：Sₙ = n⁄2 (u₁ + uₙ) 或 Sₙ = n⁄2 [2u₁ + (n−1)d]。练习 3H 常常测试在这两种形式之间切换的能力。例如，已知总和求项数时，第二种形式会变成关于 n 的二次方程：Sₙ = n⁄2 [2u₁ + (n−1)d] => 2Sₙ = n[2u₁ + (n−1)d]。重新整理后通过因式分解或求根公式解出 n。务必验证 n 为正整数。
Sₙ = ½ n (u₁ + uₙ)
4. Geometric Sequences: Common Ratio and nth Term | 等比数列：公比与第 n 项
A geometric sequence has a constant ratio r between terms: uₙ = u₁ · rn−1 . Sometimes a term is given as a fraction or negative, requiring careful sign handling. Remember that r can be negative, leading to alternating signs in the sequence.
等比数列的项与项之间具有常数公比 r：uₙ = u₁ · rn−1。练习 3H 可能要求根据连续项求公比，例如，若 u₂ = 6 且 u₅ = 162，则 u₅ / u₂ = r³ = 162/6 = 27，因此 r = 3。有时某项以分数或负数给出，需谨慎处理符号。切记 r 可以为负数，这会导致数列各项符号交替变化。
5. Geometric Series: Finite and Infinite Sums | 等比级数：有限与无限求和
For a finite geometric series with r ≠ 1, Sₙ = u₁(1 − rⁿ) / (1 − r). If |r| tal distance or amount is an infinite sum. Always check the value of r before using the infinite sum formula; if |r| ≥ 1, the sum diverges, and an infinite sum cannot be found.
对于 r ≠ 1 的有限等比级数，Sₙ = u₁(1 − rⁿ) / (1 − r)。若 |r| < 1，无穷级数收敛：S∞ = u₁ / (1 − r)。练习 3H 常包含皮球弹跳或重复过程的文字题，其中总路程或总量可表示为无穷级数和。在使用无穷和公式之前，务必检查 r 的值；若 |r| ≥ 1，级数发散，则无法求无穷和。
S∞ = u₁ / (1 − r) , |r| < 1
6. Formula Summary Table | 公式总结表
Having a clear overview of the core formulas helps tackle mixed Exercise 3H questions efficiently. The table below contrasts arithmetic and geometric sequences and series, including the conditions that apply. Keeping this reference in mind allows you to quickly identify the correct tool for each problem.
清晰梳理核心公式有助于高效应对练习 3H 中的混合题目。下表对比了等差数列、等比数列及其级数，并标明了适用条件。熟记此参考表可帮助你快速为每个问题找到正确的工具。
| Property | Arithmetic | Geometric |
|---|---|---|
| nth term | uₙ = u₁ + (n−1)d | uₙ = u₁ · rn−1 |
| Sum of first n terms | Sₙ = n/2 (u₁ + uₙ) = n/2 [2u₁ + (n−1)d] | Sₙ = u₁(1 − rⁿ)/(1 − r), r ≠ 1 |
| Infinite sum | Only finite sums exist | S∞ = u₁/(1 − r), |r| < 1 |
| Common difference / ratio | d = uₙ₊₁ − uₙ | r = uₙ₊₁ / uₙ |
7. Applications: Real-World Sequence Problems | 应用：实际数列问题
e contexts. In Exercise 3H, you might find problems on compound interest (A = P(1 + r/100)n ⁿ). These are geometric sequences in disguise. Identify u₁ (initial quantity), r (growth or decay factor), and n (number of periods). For sums, be clear whether you need the final amount after n periods or the total accumulated over several periods.
IB 数学非常注重对现实生活情境的建模。练习 3H 中可能会遇到关于复利 (A = P(1 + r/100)n)、人口增长 (Pₙ = P₀(1 + 增长率)ⁿ) 或折旧 (V = V₀(1 − 折旧率)ⁿ) 的问题。这些本质上都是等比数列。要明确首项 u₁（初始量）、公比 r（增长或衰减因子）以及期数 n。对于求和，需明确是求 n 期后的最终值，还是若干期的累积总量。
A typical example: A car depreciates by 15% each year. Its initial value is $20 000. What is its value after 5 years? Here r = 0.85, so u₆ = 20000 × (0.85)5 ≈ $8874. Exercise 3H may also ask for total depreciation over time, requiring a sum of terms.
一个典型例子：一辆汽车每年贬值 15%，初始价值为 20000 美元。5 年后的价值是多少？这里 r = 0.85，因此 u₆ = 20000 × (0.85)5 ≈ 8874 美元。练习 3H 还可能要求计算一段时间内的总折旧额，这需要求各项之和。
8. Sigma Notation and Series Manipulation | 求和符号与级数操作
m in sigma form, e.g., 2 + 6 + 18 + ... + 1458. Spotting the pattern as geometric with r=3 helps write Σ 2·3k−1 from k=1 to n, and solve 2·3n−1=1458 to find the upper limit.
求和符号 ( Σ ) 是级数的一种简洁表示方式。练习 3H 可能会给出形如 Σ (2k + 1) 从 k=1 到 20 的求和式。求解时，要识别出该数列为 3, 5, 7, ...，即等差数列，首项 u₁=3，公差 d=2，项数 n=20。于是 S₂₀ = 20/2 (2·3 + 19·2) = 10(6+38) = 440。另一类挑战是将级数写成 sigma 形式，例如 2 + 6 + 18 + ... + 1458。若能发现这是 r=3 的等比数列，就可表示为 Σ 2·3k−1 从 k=1 到 n，然后解 2·3n−1=1458 求得上限。
9. Using Technology: GDC Tips for Sequences | 技术使用：图形计算器数列技巧
IB exams allow a graphic display calculator (GDC), and Exercise 3H includes problems that are much faster with GDC. Use the sequence mode to generate terms: on TI-nspire, type seq(u₁+(n−1)d, n, 1, 10) for the first 10 terms. For sums, use sum(seq(...)) or the built-in summation template. When solving for n or r, you can graph the left and right sides of an equation and find the intersection. Always write the setup steps in your working, even when the GDC gives the answer.
IB 考试允许使用图形计算器 (GDC)，练习 3H 中的某些题目借助 GDC 可事半功倍。利用数列模式生成各项：在 TI-nspire 上，输入 seq(u₁+(n−1)d, n, 1, 10) 可得到前 10 项。求和时，使用 sum(seq(...)) 或内置的求和模板。在求解 n 或 r 时，可绘制等式左右两边的图像并求交点。即使由 GDC 得出答案，解题步骤中也要写出建立方程的过程。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Several pitfalls frequently catch students in Exercise 3H. First, confusing the 'n' but sometimes students use n=20 incorrectly in the difference. Second, forgetting to check |r| < 1 before using the infinite sum formula, which
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图