Chapter 6

Sequences and Linear Graphs

The $n$ th term of a sequence, and the equation $y = m x + c$ that turns a rule into a line you can read gradient and intercept straight off.

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

6.1 The $n$th term

Finding the rule for a linear sequence

1Find the common difference $d$
2Write $d n$ as a first attempt
3Compare it with the sequence and add or subtract the gap
4Test on the first AND a later term

ตัวอย่าง

Find the $n$ th term of $7, 11, 15, 19, \dots$ and the $80$ th term.

วิธีทำ

1.Common difference $4$ , so start with $4 n$ : $4, 8, 12, 16$
2.The sequence is $3$ more each time
3.Rule: $4 n + 3$ . Test $n = 4$ : $19$ — correct
4. $n = 80$ : $4 (80) + 3$

ตอบ $4 n + 3$ ; the $80$ th term is $323$

เคล็ดลับ

A sequence that goes DOWN has a negative coefficient. For $20, 17, 14, \dots$ the difference is $- 3$ , giving $- 3 n + 23$ — and testing $n = 1$ is what confirms the constant.

ทำไมถึงเป็นแบบนั้น

The $n$ th term matters because it answers questions listing cannot. "Is $138$ in this sequence?" becomes "does $4 n + 3 = 138$ have a whole-number solution?" — and $n = 33.75$ says no, in one line.

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