Sequences
Introduced| English | Meaning |
|---|---|
| sequence/ˈsiːkwəns/ | sequence |
| term-to-term rule/tɜːm tə tɜːm ruːl/ | A rule giving each term from the one before, such as "add 3". |
| common difference/ˈkɒmən ˈdɪfrəns/ | The constant amount added to each term to obtain the next term in an arithmetic sequence. |
| nth term/ˌenˌtiːˈeɪtʃ tɜːm/ | nth term |
| quadratic sequence/kwɒˈdrætɪk ˈsiːkwəns/ | quadratic sequence |
| geometric sequence/ˌdʒiːəʊˈmetrɪk ˈsiːkwəns/ | A sequence in which each term is obtained by multiplying the previous term by a fixed common ratio. |
| common ratio/ˈkɒmən ˈreɪʃɪəʊ/ | The factor multiplying each term to obtain the next term in a geometric sequence. |
Predicting the future
- A sunflower adds a ring of petals following the Fibonacci pattern: 1, 1, 2, 3, 5, 8, 13, …
- Sequences 数列 are patterns in numbers. Once you know the rule, you can predict any term — even the millionth.
Term-to-term rule 递推规则
- The simplest sequences add (or subtract) a fixed amount each time: $2, 5, 8, 11, \dots$ goes up by $3$.
- This is the term-to-term rule (the common difference 公差).
Romanesco broccoli: self-similar spirals form a natural number pattern
Sequences
uₙ = a + (n − 1)d
An arithmetic sequence adds the same step each time; the sum grows steadily.
The nth term 通项 (position-to-term rule)
- For a linear sequence (constant difference $d$), the nth term is $dn + c$.
- $2, 5, 8, 11, \dots$: difference $d = 3$, first term $= 2$, so $3n - 1$.
- Check: $n=1 \Rightarrow 3(1)-1 = 2$ ✓; $\;n=4 \Rightarrow 3(4)-1 = 11$ ✓.
Finding c. The sequence starts at $2$ when $n=1$: $3(1) + c = 2 \Rightarrow c = -1$. So the nth term is $3n - 1$.
The nth term of 2, 5, 8, 11, … is 3n − 1. What is the 10th term?
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3 × 10 − 1 = 30 − 1 = 29.
For a linear sequence going up by 3 each time, the nth term contains:
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A constant difference of 3 means the nth term is 3n adjusted by a constant.
The nth term of 4, 7, 10, 13, … is 3n + ______.
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3(1) + c = 4 → c = 1. The nth term is 3n + 1.
Simple quadratic sequences 二次数列
- When the first differences change but the second differences are constant, it's a quadratic sequence.
- $2, 5, 10, 17, \dots$: first differences $3, 5, 7$; second differences $2, 2$.
- The $n^2$ coefficient is half the second difference: $\dfrac{2}{2} = 1$, so the $n^2$ term is $n^2$.
- Adjust: $n^2 + 1$ gives $2, 5, 10, 17$. ✓
Number lines show patterns too: in a quadratic sequence, the gaps between terms grow steadily.
Half the second difference. The $n^2$ coefficient is $\dfrac{\text{second difference}}{2}$, not the second difference itself. For $2, 5, 10, 17$ the second difference is $2$, giving $n^2$ (not $2n^2$).
The nth term of 2, 5, 10, 17, … is n² + 1. What is the 5th term?
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5² + 1 = 25 + 1 = 26.
In a quadratic sequence, the second differences are constant.
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The first differences change steadily, but the second differences (differences of differences) are constant.
Geometric sequences 等比数列
- Each term is multiplied by a fixed ratio: $3, 6, 12, 24, \dots$ (multiply by 2 each time).
- The nth term is $a \times r^{n-1}$, where $a$ is the first term and $r$ is the common ratio 公比.
- $3 \times 2^{n-1}$: $\;n=1 \Rightarrow 3$, $\;n=4 \Rightarrow 3 \times 8 = 24$. ✓
The nth term of 3, 6, 12, 24, … is 3 × 2^(n−1). What is the 6th term?
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3 × 2⁵ = 3 × 32 = 96.
Find the fifth term of uₙ = n³ + 2.
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5³ + 2 = 125 + 2 = 127.
Cubes and a term-number check
- Core includes simple cubic sequences. $3,10,29,66$ are 2 more than $1^3,2^3,3^3,4^3$, so $u_n=n^3+2$. The fifth term is $u_5=125+2=127$.
- For $u_n=4n+3$, testing whether 100 is a term gives $4n+3=100$, so $n=24.25$. A term number must be a positive integer, so 100 is not in this sequence.
You've got it
- linear sequence: nth term $= dn + c$ (constant difference $d$) → $2,5,8,11 \to 3n-1$
- simple quadratic: second difference $= 2 \times$ the $n^2$ coefficient → $2,5,10,17 \to n^2 + 1$
- geometric: nth term $= a \times r^{n-1}$ (constant ratio $r$) → $3,6,12,24 \to 3 \times 2^{n-1}$