Arithmetic and Geometric Sequences
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| arithmetic sequence/əˈrɪθmətɪk ˈsiːkwəns/ | cấp số cộng |
| common difference/ˈkɒmən ˈdɪfrəns/ | công sai |
| geometric sequence/ˌdʒiːəʊˈmetrɪk ˈsiːkwəns/ | cấp số nhân |
| common ratio/ˈkɒmən ˈreɪʃɪəʊ/ | công bội |
| explicit/ekˈsplɪsɪt/ | hiển thị |
| recursive/rɪˈkɜːsɪv/ | lặp lại |
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.
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.
An arithmetic sequence has a constant…
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Each term is the previous one plus a fixed common difference — that steady addition makes it arithmetic.
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.
Step through a sequence term by term
Change the common ratio and watch a geometric sequence bend upward far faster than any arithmetic one.
A geometric sequence has a constant…
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Each term is the previous one times a fixed common ratio — steady multiplication makes it geometric.
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.
A ____ rule defines each term using the term before it.
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A recursive rule like $a_n = a_{n-1} + 3$ builds from the previous term; an explicit rule jumps straight to $a_n$.
A geometric sequence starts $3, 6, 12, 24, \dots$. What is the 5th term?
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The common ratio is $2$, so the 5th term is $24 \times 2 = 48$.
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.
Select all true statements about these sequences.
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Sequences can start at any value. The other three describe the add-vs-multiply difference correctly.
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.
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: $a_6 = 4 \cdot 3^5 = 4 \cdot 243 = 972$.
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.
Write from a fifth term of 80
- A geometric sequence starts 5, 10, 20, 40.
- Find the fifth term.
A geometric sequence starts 5, 10, 20, 40. What is the fifth term?
40 times 2 is 80.
What is the fifth term?
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One more doubling gives 80. Other options — 48 (the lesson later term); 40 (the fourth term).