Linear Number Patterns
Master finding the general term of a linear pattern, solving for n, finding terms from given conditions, word problems, and the connection to straight-line graphs — with full worked examples and exam strategies.
Number Patterns
Table of Contents
Number Patterns: Linear Sequences & the General Term#
A number pattern is a list of numbers that follows a rule. In Grade 10, we focus on linear patterns — where the difference between consecutive terms is constant. If you can find the rule, you can predict any term in the sequence.
The Key Concepts#
Common Difference ($d$)#
$$d = T_2 - T_1 = T_3 - T_2 = \ldots$$If $d$ is the same throughout, the pattern is linear (arithmetic).
The General Term#
$$T_n = a + (n - 1)d \quad \text{or equivalently} \quad T_n = dn + (a - d)$$| Symbol | Meaning |
|---|---|
| $T_n$ | The $n$-th term |
| $a$ ($= T_1$) | The first term |
| $d$ | The common difference |
| $n$ | The position (term number) |
Worked Example#
Pattern: $5;\, 9;\, 13;\, 17;\, \ldots$
Step 1 — Find $d$: $d = 9 - 5 = 4$
Step 2 — Write the general term: $T_n = 5 + (n-1)(4) = 5 + 4n - 4 = 4n + 1$
Step 3 — Check: $T_1 = 4(1) + 1 = 5$ ✓, $T_3 = 4(3) + 1 = 13$ ✓
Step 4 — Find the 50th term: $T_{50} = 4(50) + 1 = 201$
Solving for $n$ (“Which term equals…?”)#
Which term of the pattern equals 81?
$4n + 1 = 81 \Rightarrow 4n = 80 \Rightarrow n = 20$
The 20th term equals 81.
The Connection to Functions#
A linear pattern is just a straight line in disguise. If you plot $T_n$ against $n$, you get a straight line with gradient $d$ and y-intercept $(a - d)$.
Deep Dive#
- Linear Patterns: Full Worked Examples — finding the general term, solving for $n$, pattern problems with diagrams, and linking patterns to graphs
🚨 Common Mistakes#
- Forgetting to check the common difference is constant: If $d$ changes between pairs, it’s NOT a linear pattern.
- Off-by-one errors: $T_n = a + (n-1)d$, not $T_n = a + nd$. The $(n-1)$ is critical.
- Not checking the formula: After finding $T_n$, substitute $n = 1, 2, 3$ to verify you get the original sequence.
- $n$ must be a positive integer: If solving $T_n = k$ gives $n = 7.5$, then no term equals $k$ exactly.
🔗 Related Grade 10 topics:
- Functions: Straight Line — $T_n = dn + c$ is the same as $y = mx + c$
- Equations — solving for $n$ is a linear equation
📌 Where this leads in Grade 11: Quadratic Patterns — second differences and $T_n = an^2 + bn + c$
⏮️ Exponents | 🏠 Back to Grade 10 | ⏭️ Equations & Inequalities