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📈 等比数列

Geometric Sequences

等比数列是相邻两项之比为常数的数列。在 AMC 8 中常见于翻倍增长、折半衰减、复利计算等实际问题。

📚 3 章节 💡 5 道例题 ✏️ 8 道练习 🎯 难度：基础~中等 ⏱ 约35分钟
1
等比数列的定义 Definition of Geometric Sequences
核心必考

1.1 公比 r Common Ratio

等比数列（geometric sequence）是相邻两项之比为常数的数列。这个常数比称为公比（common ratio），记为 r。

A geometric sequence has a constant ratio between consecutive terms. This constant is called the common ratio, denoted r.

📝 公比 / Common Ratio
r = a₂ ÷ a₁ = a₃ ÷ a₂ = a₄ ÷ a₃ = ...
公比可以是正数、负数或分数（但不能为0）
The ratio can be positive, negative, or a fraction (but not zero)

举例：

  • 2, 6, 18, 54, ... → r = 6÷2 = 3（增长）
  • 64, 32, 16, 8, ... → r = 32÷64 = 1/2（衰减）
  • 3, −6, 12, −24, ... → r = −6÷3 = −2（正负交替）

1.2 通项公式 General Term Formula

等比数列的通项公式：

The formula for the nth term:

📝 等比数列通项公式 / General Term
an = a₁ × r(n−1)
其中 a₁ 是首项，r 是公比，n 是项数
where a₁ = first term, r = common ratio, n = term number
💡 邓老师提示：等比数列中，每一项都等于前一项乘以公比 r。判断是否为等比数列，就看相邻两项的比值是否相同。
In a geometric sequence, each term = previous term × r. To check if a sequence is geometric, verify the ratio between consecutive terms is constant.
2
等比数列的应用 Applications of Geometric Sequences
中等高频

2.1 翻倍增长 Doubling Growth

最经典的等比数列应用之一：当一个量每次翻倍（或乘以固定倍数）增长时，形成等比数列。

One of the most classic applications: when a quantity doubles (or multiplies by a fixed factor) each time.

📝 翻倍增长 / Doubling
初始值 P，每次翻倍：
第1次：P × 2
第2次：P × 2²
第n次：P × 2n
Starting with P, after n doublings: P × 2n

举例：一张纸对折一次变2层，对折10次有多少层？

  • 层数 = 1 × 210 = 1024 层

2.2 衰减问题 Decay Problems

当一个量每次变为原来的固定比例（如减半、衰减为90%）时，也形成等比数列。

When a quantity becomes a fixed fraction of itself each time (halving, 90% remaining, etc.), it forms a geometric sequence.

📝 衰减 / Decay
初始值 P，每次变为原来的 r 倍（0 < r < 1）：
第n次后：P × rn
After n steps: P × rn (where 0 < r < 1)

举例：一个弹性球从 100 米高度落下，每次弹起高度为上次的 80%。第3次弹起多高？

  • 第3次高度 = 100 × 0.8³ = 100 × 0.512 = 51.2 米
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等比数列求和 Sum of a Geometric Sequence
核心高频

3.1 有限等比求和公式 Finite Geometric Sum Formula

📝 等比数列求和公式 / Geometric Sum Formula
Sn = a₁ × (1 − rn) ÷ (1 − r)   （r ≠ 1）
当 r = 1 时，Sn = n × a₁（所有项相同）
When r = 1, all terms are equal, so S = n × a₁
💡 邓老师提示：这个公式看起来复杂，但 AMC 8 中通常 r 是简单数（2, 3, 1/2 等），n 也不大，可以直接代入计算。
This formula looks complex, but in AMC 8, r is usually simple (2, 3, 1/2) and n is small, so just plug in.

3.2 求和公式的应用 Applying the Sum Formula

举例：求 1 + 2 + 4 + 8 + 16 + 32 的值

Example: Find 1 + 2 + 4 + 8 + 16 + 32

  • a₁ = 1, r = 2, n = 6
  • S = 1 × (1 − 26) ÷ (1 − 2) = (1 − 64) ÷ (−1) = (−63) ÷ (−1) = 63
⚠️ 注意：当 r > 1 时，(1−r) 是负数，分子 (1−rn) 也是负数，负负得正。也可以用 S = a₁(rn−1)/(r−1) 来避免负数。
When r > 1, you can also use S = a₁(rn−1)/(r−1) to avoid negative numbers.
4
例题精讲 Worked Examples
5 题含历年真题
📌 例题 1 通项公式

等比数列首项为 3，公比为 2。第 8 项是多少？A geometric sequence has first term 3 and common ratio 2. What is the 8th term?

解题思路
a₈ = 3 × 27 = 3 × 128 = 384 a₈ = 3 × 27 = 3 × 128 = 384.
📌 例题 2 等比求和

求 3 + 6 + 12 + 24 + ... + 768 的和。Find the sum 3 + 6 + 12 + 24 + ... + 768.

解题思路
a₁=3, r=2, aₙ=768。768=3×2n−1 → 2n−1=256=2⁸ → n=9
S = 3×(2⁹−1)/(2−1) = 3×(512−1)/1 = 3×511 = 1533 a₁=3, r=2, n=9. S = 3(2⁹−1)/(2−1) = 3×511 = 1533.
📌 例题 3 AMC 8 真题改编

某种细菌每 3 小时数量翻倍。开始时有 10 个细菌，9 小时后有多少个？A type of bacteria doubles every 3 hours. If there are 10 bacteria initially, how many are there after 9 hours?

解题思路
9 ÷ 3 = 3 次翻倍
细菌数 = 10 × 2³ = 10 × 8 = 80 个 9 ÷ 3 = 3 doublings. Bacteria = 10 × 2³ = 10 × 8 = 80.
📌 例题 4 衰减问题

一个球的弹性系数是 0.5（每次弹起高度为上次的一半）。从 160 米落下，第4次碰地后弹起的高度是多少？A ball bounces to half its previous height. Dropped from 160m, how high does it bounce after the 4th bounce?

解题思路
第1次弹起：160 × 0.5 = 80
第2次弹起：80 × 0.5 = 40
第3次弹起：40 × 0.5 = 20
第4次弹起：20 × 0.5 = 10 米
或直接：160 × (0.5)⁴ = 160/16 = 10 After 4th bounce: 160 × (0.5)⁴ = 160/16 = 10 meters.
📌 例题 5 综合

等比数列 1, 3, 9, 27, ... 中，第一次超过 1000 的是第几项？In the geometric sequence 1, 3, 9, 27, ..., which term first exceeds 1000?

解题思路
aₙ = 3n−1 > 1000
3⁵=243, 3⁶=729, 3⁷=2187
3⁶ = 729 < 1000, 3⁷ = 2187 > 1000
所以第 7 项第一次超过 1000 3⁶=729 < 1000, 3⁷=2187 > 1000. The 7th term first exceeds 1000.
5
巩固练习 Practice Problems
8 题提交即判

第1题 等比数列 5, 15, 45, 135, ... 的第 6 项是多少？What is the 6th term of the geometric sequence 5, 15, 45, 135, ...?

第2题 等比数列首项为 2，公比为 3。前 5 项的和是多少？A geometric sequence has first term 2 and common ratio 3. Find the sum of the first 5 terms.

第3题 一个正方形每次边长变为原来的一半。初始边长为 32，第5次变化后的边长是多少？A square's side length halves each time. Starting at 32, what is the side length after 5 changes?

第4题 等比数列的第 3 项是 12，第 6 项是 96。首项 a₁ 是多少？The 3rd term is 12 and the 6th term is 96. What is a₁?

第5题 求等比数列 1, 1/2, 1/4, 1/8, ..., 1/128 的和。Find the sum of 1, 1/2, 1/4, 1/8, ..., 1/128.

第6题 一笔钱每年增长 10%（即乘以 1.1）。100 元 3 年后变为多少元？Money grows 10% per year (multiplied by 1.1). How much is 100 yuan after 3 years?

第7题 等比数列 2, −6, 18, −54, ... 的第 7 项是多少？What is the 7th term of 2, −6, 18, −54, ...?

第8题 一只青蛙每次跳的距离是上次的 2/3。第一次跳 27 厘米，第4次跳多少厘米？A frog jumps 2/3 of its previous distance. First jump is 27cm. How far is the 4th jump?

⬅ 上一页 等差数列 / Arithmetic Sequences