Arithmetic and Geometric Sequences · 等差数列与等比数列
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| arithmetic sequence/əˈrɪθmətɪk ˈsiːkwəns/ | 等差数列 | děng chā shù liè |
| common difference/ˈkɒmən ˈdɪfrəns/ | 公差 | gōng chāi |
| geometric sequence/ˌdʒiːəʊˈmetrɪk ˈsiːkwəns/ | 等比数列 | děng bǐ shù liè |
| common ratio/ˈkɒmən ˈreɪʃɪəʊ/ | 公比 | gōng bǐ |
| explicit/ekˈsplɪsɪt/ | 显式 | xiǎn shì |
| recursive/rɪˈkɜːsɪv/ | 递归 | dì guī |
Two ways to grow a list
- Start at $2$ and keep adding $2$: you get $2, 4, 6, 8, \dots$
- Start at $1$ and keep doubling: you get $1, 2, 4, 8, \dots$
- Both are ordered lists of numbers — sequences — but they grow in completely different ways.
- One adds the same amount; the other multiplies by the same amount.
让一串数字增长的两种方式
- 从 $2$ 开始,不断加 $2$:你得到 $2, 4, 6, 8, \dots$
- 从 $1$ 开始,不断翻倍:你得到 $1, 2, 4, 8, \dots$
- 两者都是有序的数字列表——数列——但它们的增长方式完全不同。
- 一个加相同的量;另一个乘以相同的量。
Arithmetic: add the same amount
- An arithmetic sequence 等差数列 adds a fixed number to get from each term to the next.
- That fixed number is the common difference 公差.
- In $5, 8, 11, 14, \dots$ the common difference is $+3$.
- Graphed, the terms sit on a straight line — steady, linear growth.
等差:加上相同的量
- 等差数列(arithmetic sequence)从每一项到下一项都加上一个固定的数。
- 那个固定的数就是公差(common difference)。
- 在 $5, 8, 11, 14, \dots$ 中,公差是 $+3$。
- 画出来,各项落在一条直线上——稳定的、线性的增长。
An arithmetic sequence has a constant… · 等差数列有一个恒定的……
Each term is the previous one plus a fixed common difference — that steady addition makes it arithmetic. · 每一项都是前一项加上一个固定的公差——这种稳定的加法使它成为等差。
Geometric: multiply by the same amount
- A geometric sequence 等比数列 multiplies by a fixed number each step.
- That fixed number is the common ratio 公比.
- In $3, 6, 12, 24, \dots$ the common ratio is $\times 2$.
- Graphed, the terms curve upward ever more steeply.
等比:乘以相同的量
- 等比数列(geometric sequence)每一步都乘以一个固定的数。
- 那个固定的数就是公比(common ratio)。
- 在 $3, 6, 12, 24, \dots$ 中,公比是 $\times 2$。
- 画出来,各项越来越陡地向上弯。
Step through a sequence term by term · 一项一项地走过一个数列
Change the common ratio and watch a geometric sequence bend upward far faster than any arithmetic one. · 改变公比,看等比数列比任何等差数列都弯得快得多。
A geometric sequence has a constant… · 等比数列有一个恒定的……
Each term is the previous one times a fixed common ratio — steady multiplication makes it geometric. · 每一项都是前一项乘以一个固定的公比——稳定的乘法使它成为等比。
Explicit and recursive rules
- An explicit 显式 rule jumps straight to any term: arithmetic $a_n = a_1 + (n-1)d$; geometric $a_n = a_1 \cdot r^{\,n-1}$.
- A recursive 递归 rule builds each term from the one before: $a_n = a_{n-1} + d$ or $a_n = a_{n-1} \cdot r$.
- Explicit is best for finding the 100th term without listing all the others.
- Recursive mirrors how the sequence is actually generated, step by step.
显式规则与递归规则
- 显式(explicit)规则直接跳到任意一项:等差 $a_n = a_1 + (n-1)d$;等比 $a_n = a_1 \cdot r^{\,n-1}$。
- 递归(recursive)规则从前一项构建每一项:$a_n = a_{n-1} + d$ 或 $a_n = a_{n-1} \cdot r$。
- 显式最适合直接求第 100 项,而不必列出前面所有项。
- 递归则反映了数列实际一步步生成的方式。
A ____ rule defines each term using the term before it. · ____规则用前一项来定义每一项。
A recursive rule like $a_n = a_{n-1} + 3$ builds from the previous term; an explicit rule jumps straight to $a_n$. · 像 $a_n = a_{n-1} + 3$ 这样的递归规则从前一项构建;显式规则则直接跳到 $a_n$。
A geometric sequence starts $3, 6, 12, 24, \dots$. What is the 5th term? · 一个等比数列是 $3, 6, 12, 24, \dots$。第 5 项是多少?
The common ratio is $2$, so the 5th term is $24 \times 2 = 48$. · 公比是 $2$,所以第 5 项是 $24 \times 2 = 48$。
Which grows faster
- Early on, an arithmetic sequence can be ahead.
- But any geometric sequence with ratio $> 1$ eventually overtakes it — and then pulls away fast.
- Adding is linear; multiplying is exponential, and exponential always wins in the long run.
- This is the same contrast that separates linear from exponential functions.
谁增长得更快
- 早期,等差数列可能领先。
- 但任何公比 $> 1$ 的等比数列最终都会超过它——然后飞速拉开距离。
- 加法是线性的;乘法是指数的,而指数从长远看总是获胜。
- 这与区分线性函数和指数函数的对比是同一个。
Select all · 所有 true statements about these sequences. · 选出关于这些数列的所有正确说法。
Sequences can start at any value. The other three describe the add-vs-multiply difference correctly. · 数列可以从任何值开始。其余三条正确地描述了"加"与"乘"的区别。
Check whether a list adds or multiplies before naming it. $2, 4, 6, 8$ (add 2) is arithmetic, but $2, 4, 8, 16$ (times 2) is geometric — the first two terms look identical, so always test a third term.
在命名一串数字之前,先检查它是加还是乘。$2, 4, 6, 8$(加 2)是等差,而 $2, 4, 8, 16$(乘 2)是等比——头两项看起来一模一样,所以永远要测第三项。
Find the 6th term of $4, 12, 36, \dots$.
- Ratio: $12 \div 4 = 3$, so this is geometric with $r = 3$.
- Explicit rule: $a_n = 4 \cdot 3^{\,n-1}$.
- Sixth term: $a_6 = 4 \cdot 3^5 = 4 \cdot 243 = 972$.
求 $4, 12, 36, \dots$ 的第 6 项。
- 比:$12 \div 4 = 3$,所以这是公比 $r = 3$ 的等比数列。
- 显式规则:$a_n = 4 \cdot 3^{\,n-1}$。
- 第六项:$a_6 = 4 \cdot 3^5 = 4 \cdot 243 = 972$。
An arithmetic sequence adds a common difference; a geometric sequence multiplies by a common ratio. Write either with an explicit rule (jump to any term) or a recursive rule (build from the previous term). Geometric growth eventually overtakes arithmetic.
等差数列加一个公差;等比数列乘以一个公比。可以用显式规则(直接跳到任意项)或递归规则(从前一项构建)来写它。等比增长最终会超过等差。