| English | 意味 |
|---|---|
| arithmetic sequence/əˈrɪθmətɪk ˈsiːkwəns/ | 等差数列 |
| common difference/ˈkɒmən ˈdɪfrəns/ | 公差 |
| geometric sequence/ˌdʒiːəʊˈmetrɪk ˈsiːkwəns/ | 等比数列 |
| common ratio/ˈkɒmən ˈreɪʃɪəʊ/ | 公比 |
| explicit/ekˈsplɪsɪt/ | 一般項 |
| recursive/rɪˈkɜːsɪv/ | 再帰的 |
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.
等差数列には一定の…がある
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
各項は前項に一定の共通差分を足すことで得られる——この安定した加算が等差数列である。
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.
項を一つずつ順を追って確認する
公比を変化させると、等比数列は等差数列よりもはるかに速く上方に曲がるのがわかる。
等比数列には一定の…がある
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
各項は前項に一定の公比をかけることで得られる——安定した乗算が等比数列である。
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.
前項を用いて各項を定義する____ルール
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
$a_n = a_{n-1} + 3$のような再帰的ルールは前項から積み上げていく；明示的ルールは$a_n$へ直接飛躍する。
ある等比数列の初項は$3, 6, 12, 24, \dots$である。第5項はいくつか？
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
公比が$2$なので、第5項は$24 \times 2 = 48$である。
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.
これらの数列に関する真の文をすべて選択せよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
数列はどのような値でも始められる。残りの3つは加算と乗算の違いを正しく説明している。
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.