📚 Geometric Sequences and Series | 等比数列与级数
A geometric sequence, also known as a geometric progression, is a fundamental concept in IB Mathematics. It describes a list of numbers where each term is obtained by multiplying the previous term by a fixed, non-zero constant called the common ratio. Understanding geometric sequences and their associated series is essential for modelling exponential growth, compound interest, and many other real-world phenomena.
等比数列（也称几何数列）是 IB 数学中的一个基本概念。它描述的是一组数字，其中每一项都是前一项乘以一个固定且非零的常数（称为公比）得到的。理解等比数列及其级数对于模拟指数增长、复利和许多其他现实世界现象至关重要。
1. Introduction to Geometric Sequences | 等比数列简介
A geometric sequence is an ordered list of numbers in which the ratio of any term to its immediate predecessor is constant. This fixed ratio is the defining characteristic of the sequence. For instance, the sequence 3, 12, 48, 192, ... is geometric, because each term after the first is 4 times the previous term.
等比数列是一个有序的数字列表，其中任意一项与其直接前一项的比值是恒定的。这个固定的比值是该数列的定义性特征。例如，数列 3, 12, 48, 192, ... 是等比数列，因为从第二项起每一项都是前一项的 4 倍。
Geometric sequences can be either finite (having a limited number of terms) or infinite (continuing indefinitely). The first term is usually denoted by u1 and the common ratio by r . It is crucial that r ≠ 0 ; otherwise, all terms after the first would be zero, and the sequence would lose its multiplicative pattern.
等比数列可以是有限的（项数有限）或无限的（无限延续）。首项通常记作 u1 ，公比记作 r 。至关重要的是 r ≠ 0 ；否则，首项之后的所有项都将为零，数列的乘积模式也会消失。
2. The Common Ratio | 公比
The common ratio r is calculated by dividing any term by the term that precedes it, provided the preceding term is not zero. Mathematically, r = un+1⁄un for any positive integer n . This ratio remains the same throughout the sequence.
公比 r 的计算方法是：用任何一项除以其前一项（前提是前一项不为零）。数学上，对于任意正整数 n ，有 r = un+1⁄un 。这个比值在整个数列中保持不变。
The value of r determines the behaviour of the sequence. If r > 1 , the terms grow in magnitude; if 0 < r < 1 , the terms decrease towards zero. When r < 0 , the terms alternate in sign, creating an oscillating pattern. If r = 1 , the sequence is constant.
r 的值决定了数列的行为。如果 r > 1 ，各项的绝对值增大；如果 0 < r < 1 ，各项递减趋向于零。当 r < 0 时，各项正负交替，形成振荡模式。如果 r = 1 ，数列为常数数列。
3. General Term Formula | 通项公式
The n -th term of a geometric sequence is given by the general term formula:
等比数列的第 n 项由通项公式给出：
un = u1 · rn−1
Here, u1 is the first term, r is the common ratio, and n is the term number. This formula allows you to find any term directly without having to calculate all the preceding ones.
其中 u1 是首项， r 是公比， n 是项数。这个公式能让你直接求出任意一项，而无需计算之前的所有项。
For example, if a sequence has u1 = 7 and r = 3 , the 5th term is u5 = 7 × 34 = 7 × 81 = 567 . The formula is valid for any real value of r , provided r ≠ 0 .
例如，若数列的首项为 7，公比为 3，则第 5 项为 u5 = 7 × 34 = 7 × 81 = 567 。只要 r ≠ 0 ，该公式对任何实数值 r 均成立。
4. Finding a Specific Term | 求特定项
In many IB problems, you are given two terms of a geometric sequence and are asked to find the common ratio or another term. Suppose you know u3 = 18 and u6 = 486 . Since u6 = u3 · r3 , we have 486 = 18 · r3 , so r3 = 27 , giving r = 3 . Then u1 can be found using u3 = u1 · r2 .
在许多 IB 题目中，会给出等比数列的两项，要求求出公比或另一项。假设已知 u3 = 18 和 u6 = 486 。因为 u6 = u3 · r3 ，有 486 = 18 · r3 ，所以 r3 = 27 ，得 r = 3 。然后可用 u3 = u1 · r2 求出 u1 。
This technique of using the general term to set up equations is powerful. It can also involve logarithms when the unknown is in the exponent, which is a common skill tested in IB Mathematics: Analysis and approaches.
利用通项公式建立方程的方法非常有效。当未知量出现在指数上时，还可能涉及对数运算，这是 IB 数学“分析与方法”课程中常见的考查技能。
5. Introduction to Geometric Series | 等比级数简介
A geometric series is the sum of the terms of a geometric sequence. If the sequence is u1, u1 r, u1 r2, … , then the corresponding series is u1 + u1 r + u1 r2 + … . We distinguish between a finite geometric series (summing a fixed number of terms) and an infinite geometric series.
等比级数是等比数列各项之和。若数列为 u1, u1 r, u1 r2, … ，则对应的级数为 u1 + u1 r + u1 r2 + … 。我们区分有限等比级数（对固定项数求和）和无穷等比级数。
The notation Sn is used to represent the sum of the first n terms. Working with series often requires careful handling of the common ratio, especially when it is negative or a fraction.
符号 Sn 用于表示前 n 项之和。处理级数时往往需要谨慎对待公比，尤其是当公比为负数或分数时。
6. Sum of a Finite Geometric Series | 有限项等比级数的和
For a finite geometric series with first term u1 , common ratio r ≠ 1 , and n terms, the sum is given by:
对于首项为 u1 、公比 r ≠ 1 、项数为 n 的有限等比级数，其求和公式为：
Sn = u1 × (1 − rn) / (1 − r)
Equivalently, you can use Sn = u1 (rn − 1) / (r − 1) . Both forms are acceptable, but the first is often preferred when r < 1 to keep the numerator positive.
等效也可使用 Sn = u1 (rn − 1) / (r − 1) 。两种形式均可接受，但当 r < 1 时常用第一个公式，以保持分子为正。
For example, sum the first 8 terms of the series 5 + 15 + 45 + ... . Here u1 = 5, r = 3 . Then S8 = 5(38 − 1)/(3 − 1) = 5(6561 − 1)/2 = 5 × 6560 / 2 = 16400 .
例如，求级数 5 + 15 + 45 + ... 的前 8 项和。这里 u1 = 5, r = 3 。则 S8 = 5(38 − 1)/(3 − 1) = 5(6561 − 1)/2 = 5 × 6560 / 2 = 16400 。
7. Sum of an Infinite Geometric Series | 无穷等比级数的和
When the common ratio satisfies |r| < 1 , the terms of an infinite geometric series become smaller in magnitude, and the series converges to a finite sum. This sum is given by:
当公比满足 |r| < 1 时，无穷等比级数的各项绝对值逐渐减小，级数收敛于一个有限和。该和由下式给出：
S∞ = u1 / (1 − r)
This formula arises by taking the limit of Sn as n approaches infinity; since rn → 0 when |r| < 1 , the term rn disappears. The sum to infinity represents the total sum of all terms in the sequence.
此公式是通过取 n 趋近于无穷时 Sn 的极限得到的；由于当 |r| < 1 时 rn → 0 ， rn 项消失。无穷和表示数列所有项的总和。
For example, the series 8 + 4 + 2 + 1 + ½ + ... has u1 = 8 and r = ½ . Its sum to infinity is S∞ = 8 / (1 − ½) = 8 / (½) = 16 . This is a classic result that illustrates convergence.
例如，级数 8 + 4 + 2 + 1 + ½ + ... 的 u1 = 8 ， r = ½ 。其无穷和为 S∞ = 8 / (1 − ½) = 8 / (½) = 16 。这一经典结果很好地说明了收敛性。
8. Convergence and Divergence | 收敛与发散
An infinite geometric series converges if and only if the absolute value of the common ratio is less than 1, i.e., |r| < 1 . In that case, the series approaches a finite sum as the number of terms increases indefinitely.
无穷等比级数收敛当且仅当公比的绝对值小于 1，即 |r| < 1 。此时，随着项数无限增加，级数趋近于一个有限和。
If |r| ≥ 1 and the terms are non-zero, the series diverges; the sum does not approach a finite limit. For r = 1 , the sum of the first n terms is n × u1 , which grows without bound. For r = −1 , the partial sums oscillate between 0 and u1 , also diverging.
如果 |r| ≥ 1 且各项非零，级数发散；其和不趋近于有限极限。对于 r = 1 ，前 n 项和为 n × u1 ，无限增长。对于 r = −1 ，部分和在 0 和 u1 之间振荡，同样发散。
In IB examinations, you are often asked to find the range of values of x for which an infinite geometric series converges, and then compute its sum in terms of x .
在 IB 考试中，常要求找出使无穷等比级数收敛的 x 取值范围，然后用 x 表示其和。
9. Geometric Mean | 等比中项
The geometric mean of two positive numbers a and b is √ab . In a geometric sequence, any term (except the first and last) is the geometric mean of its two neighbouring terms. For three consecutive terms uk-1, uk, uk+1 , we have uk2 = uk-1 · uk+1 .
两个正数 a 和 b 的等比中项为 √ab 。在等比数列中，任意一项（首项和末项除外）是其相邻两项的等比中项。对于三个连续项 uk-1, uk, uk+1 ，有 uk2 = uk-1 · uk+1 。
This property can be used to find missing terms in a geometric sequence or to construct a geometric progression between two given numbers. It is analogous to the arithmetic mean in arithmetic sequences, but involves multiplication instead of addition.
这一性质可用于求等比数列中的缺失项，或在两个给定数之间构造等比数列。它与等差数列中的等差中项类似，但涉及乘法而非加法。
10. Applications: Compound Interest | 应用：复利
Geometric sequences naturally model compound interest. If a principal amount P is invested at an annual interest rate of i compounded yearly, the amount after n years is An = P (1 + i)n . This is a geometric sequence with first term A0 = P and common ratio (1 + i) .
等比数列可以自然地模拟复利。如果本金 P 以年利率 i 每年复利一次，则 n 年后的金额为 An = P (1 + i)n 。这是一个首项为 A0 = P 、公比为 (1 + i) 的等比数列。
The total amount after a series of regular deposits also leads to geometric series. For instance, the future value of an annuity can be expressed as the sum of a finite geometric series, linking financial mathematics directly to the theory of geometric progressions.
一系列定期存款后的总金额也会导出等比级数。例如，年金的未来值可以表示为有限等比级数的和，从而将金融数学与等比级数理论直接联系起来。
11. Applications: Exponential Growth and Decay | 应用：指数增长与衰减
Many natural and social phenomena follow exponential patterns that can be described by geometric sequences. Population growth, radioactive decay, depreciation of assets, and the spread of a virus often satisfy recurrence relations of the form un+1 = un · r .
许多自然和社会现象遵循可用等比数列描述的指数模式。人口增长、放射性衰变、资产折旧和病毒的传播往往满足形如 un+1 = un · r 的递推关系。
For example, if a car loses 15% of its value each year, its value forms a geometric sequence with r = 0.85 . After 5 years, its value is V5 = V0 (0.85)5 . The sum of the values over time might also be interpreted as a series if needed.
例如，如果一辆汽车每年贬值 15%，其价值形成一个公比 r = 0.85 的等比数列。5 年后，其价值为 V5 = V0 (0.85)5 。如果有需要，随时间变化的价值总和也可以解释为一个级数。
Recognising a geometric sequence in context allows students to apply formulas for term calculation and summation to solve real-world problems—a key objective of the IB curriculum.
在具体情境中识别出等比数列，能让学生应用项计算和求和的公式解决现实问题，这也是 IB 课程的一个核心目标。
12. Sigma Notation and Series Summation | 西格玛符号与求和
The sum of a geometric series is often expressed compactly using sigma notation: ∑k=0n-1 u1 rk or ∑k=1n u1 rk-1 . Interpreting this notation and translating it into the sum formula is a vital skill for IB Mathematics.
等比级数的求和通常用西格玛符号简洁地表示为： ∑k=0n-1 u1 rk 或 ∑k=1n u1 rk-1 。读懂这一符号并将其转化为求和公式是 IB 数学的一项重要技能。
For example, evaluate ∑k=14 2 · 5k-1 . Here u1 = 2, r = 5, n = 4 . So the sum is S4 = 2(54 − 1)/(5 − 1) = 2(625 − 1)/4 = 2 × 624 / 4 = 312 .
例如，计算 ∑k=14 2 · 5k-1 。这里 u1 = 2, r = 5, n = 4 。因此和为 S4 = 2(54 − 1)/(5 − 1) = 2(625 − 1)/4 = 2 × 624 / 4 = 312 。
When the series is infinite and convergent, sigma notation is also used: ∑k=0∞ u1 rk = u1⁄1-r , provided |r| < 1 . This compact notation appears frequently in IB exam questions.
当级数为无穷且收敛时，也使用西格玛符号： ∑k=0∞ u1 rk = u1⁄1-r ，前提是 |r| < 1 。这一简洁符号在 IB 考试题目中频繁出现。
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