Sequences

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

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: $n=1 \Rightarrow 3(1)-1 = 2$ ✓; $\;n=4 \Rightarrow 3(4)-1 = 11$ ✓.

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.

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.
• $3 \times 2^{n-1}$: $\;n=1 \Rightarrow 3$, $\;n=4 \Rightarrow 3 \times 8 = 24$. ✓

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.

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