Sequences, Coordinates and Graphs

Patterns in numbers, the rules behind them, and how a rule becomes a straight line on a grid — the first place algebra and geometry meet.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.1 บทที่ 8 กราฟและความสัมพันธ์เชิงเส้น
Common Core (US): 6.EE, 8.F
Cambridge: Lower Secondary Stage 7 — Algebra: sequences, functions and graphs

7.1 Sequences and their rules

นิยาม

Term-to-term rule

Says how to get from one term to the next: "add $4$ ". Simple to spot, but useless for finding the $100$ th term without listing all the others.

นิยาม

Position-to-term rule

Gives any term directly from its position $n$ : the $n$ th term of $3, 7, 11, 15, \dots$ is $4 n - 1$ . This is the rule worth finding.

Finding the $n$ th term of a linear sequence

1Find the common difference $d$ between consecutive terms
2Start with $d n$ — the sequence $d, 2 d, 3 d, \dots$
3Compare with the actual sequence and add or subtract the gap
4Test on the first and third terms before trusting it

ตัวอย่าง

Find the $n$ th term of $5, 9, 13, 17, \dots$ and use it to find the $50$ th term.

วิธีทำ

1.Common difference is $4$ , so start with $4 n$ : that gives $4, 8, 12, 16$
2.The sequence is $1$ more each time, so the rule is $4 n + 1$
3.Test: $n = 3$ gives $13$ , which matches
4. $n = 50$ : $4 (50) + 1$

ตอบ $n$ th term $= 4 n + 1$ ; the $50$ th term is $201$

ข้อควรระวัง

The common difference is the coefficient of $n$ , not the first term. A sequence starting at $5$ and going up in $4$ s has rule $4 n + 1$ , not $5 n$ — a mistake that is easy to catch by testing $n = 1$ .

📖 2 more sections of this chapter are being checked

The whole chapter is written; we read it back section by section before releasing it, because maths nobody has checked should not be on a public page. What is open is complete — nothing here is abridged.

Read it — now practise it

Generate a worksheet on this topic, or sit a timed mock paper.

Build a worksheet Practice arena