This document discusses linear and quadratic number patterns. It begins by reviewing linear patterns with terms of the form Tn = bn + c. It then introduces quadratic patterns, which have terms of the form Tn = an^2 + bn + c. An example quadratic pattern of Tn = n^2 + 1 is generated to show the constant second difference of 2a. It explains that while linear patterns have a constant first difference, quadratic patterns have a constant second difference. The document then derives the expressions for determining the nth term of any given quadratic pattern from the coefficients a, b, and c.

1. NUMBER PATTERNS
GRADE 11

2. REVISION OF LINEAR NUMBER PATTERNS
In Grade 10 we dealt with linear number patterns having a general
term of the form 𝑇𝑛 = 𝑏𝑛 + 𝑐.
EXAMPLE
Consider the following linear number pattern: 2 ; 7 ; 12 ; 17 ; ………….
(a) Determine the nth term and hence the 199th term.
(b) Which term of the number pattern is equal to 497?

4. QUADRATIC NUMBER PATTERNS
We will now focus on quadratic number patterns with general terms of
the form:
𝑇𝑛 = 𝑎𝑛2 + 𝑏𝑛 + 𝑐

5. Introduction
Consider, for example, the formula 𝑇𝑛 = 𝑛2 + 1.
We can generate the terms of the number sequence as follows:
𝑇1 = 1 2
+ 1 = 2
𝑇2 = 2 2 + 1 = 5
𝑇3 = 3 2 + 1 = 10
𝑇4 = 4 2 + 1 = 17
𝑇5 = 5 2 + 1 = 26
If we now consider the pattern 2; 5; 10; 17; 26; ………, we can investigate some
interesting properties of this quadratic number pattern.

6. Introduction
Number patterns with a constant second differences are called quadratic
number patterns.
It is clear that this number pattern does not have a constant first difference as
with linear number patterns.
However… it does have a constant second difference.

7. The 𝑛𝑡ℎ term of Quadratic Sequence
The question now arises as to how we would determine the general
term of any given quadratic number pattern.
Suppose that the general term of a particular quadratic number pattern
is given by: 𝑇𝑛 = 𝑎𝑛2 + 𝑏𝑛 + 𝑐
The terms of the number pattern would then be:
𝑇1 = 𝑎 1 2 + 𝑏 1 + 𝑐 = 𝑎 + 𝑏 + 𝑐
𝑇2 = 𝑎 2 2 + 𝑏 2 + 𝑐 = 4𝑎 + 2𝑏 + 𝑐
𝑇3 = 𝑎 3 2 + 𝑏 3 + 𝑐 = 9𝑎 + 3𝑏 + 𝑐
𝑇4 = 𝑎 4 2 + 𝑏 4 + 𝑐 = 16𝑎 + 4𝑏 + 𝑐

8. The 𝑛𝑡ℎ term of Quadratic Sequence
You will notice that the constant second difference is given by the expression 2a
The first term in the first difference row is given by 3𝑎 + 𝑏
… and the first term is given by 𝑎 + 𝑏 + 𝑐

9. The 𝑛𝑡ℎ term of Quadratic Sequence

10. The 𝑛𝑡ℎ term of Quadratic Sequence

11. Example 1

12. Example 2