Quadratic Number Patterns

A quadratic pattern has a constant second difference. Grade 12 questions often mix quadratic and linear patterns, especially by asking about the first differences, which form a linear pattern of their own.

Finding the general term of a quadratic pattern

If the second difference is constant, $T_{n} = a n^{2} + bn + c$ , where:

$2 a$ = second difference
$3 a + b$ = first of the first differences
$a + b + c$ = first term

Example. $3; 8; 15; 24; \dots$

First differences: $5; 7; 9$ . Second difference: $2$ .

$2 a = 2 \Rightarrow a = 13 + b = 5 \Rightarrow b = 21 + 2 + c = 3 \Rightarrow c = 0$

$T_{n} = n^{2} + 2 n$

Which term? $n^{2} + 2 n = 255 \Rightarrow (n - 15) (n + 17) = 0 \Rightarrow n = 15$ .

Always check your formula with $T_{2}$ or $T_{3}$ .

First differences as a linear pattern

The first differences of a quadratic pattern form an arithmetic (linear) sequence.

Example. $- 1; 2; 7; 14; \dots$ has first differences $3; 5; 7; \dots$

The first difference between $T_{n}$ and $T_{n + 1}$ is $3 + 2 (n - 1) = 2 n + 1$ .

Which two terms differ by 81? $2 n + 1 = 81 \Rightarrow n = 40$ , so between $T_{40}$ and $T_{41}$ .

Sum of first differences. $T_{n} = T_{1} + (sum of the first n - 1 differences)$ . This links quadratic patterns to arithmetic series.

Recursive patterns are not examined explicitly.

Key terms

Quadratic pattern: A pattern with a constant second difference; $T_{n} = a n^{2} + bn + c$ .
First differences: The differences between consecutive terms.
Second difference: The difference between consecutive first differences.

Common mistakes

Using the formula for a linear pattern when the first differences are not constant.
Writing $a$ = second difference instead of half of it.
Mixing up the position of a first difference with the term numbers.
Keeping a negative or non-integer $n$ .

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

5 marks total

routine

Given the quadratic pattern $3; 8; 15; 24; \dots$

(a) Determine $T_{n}$ . (3)

(b) Which term of the pattern is equal to 255? (2)

Worked solution

Hidden - try the question first

Solution

(a) Second difference 2, so $a = 1$
$3 a + b = 5 \Rightarrow b = 2$ ; $a + b + c = 3 \Rightarrow c = 0$
$T_{n} = n^{2} + 2 n$
(b) $n^{2} + 2 n - 255 = 0 \Rightarrow (n - 15) (n + 17) = 0$
$n = 15$

Question 2

5 marks total

routine

Determine $T_{30}$ of the pattern $4; 7; 12; 19; \dots$

Worked solution

Hidden - try the question first

Solution

First differences $a = 1$ , second difference 2
$a = 1$ , $b = 0$
$c = 3$
$T_{n} = n^{2} + 3$
$T_{30} = 903$

Question 3

6 marks total

complex

Given the quadratic pattern $- 1; 2; 7; 14; \dots$

(a) Determine $T_{n}$ . (3)

(b) Determine the general term of the first differences. (1)

Worked solution

Hidden - try the question first

Solution

(a) $a = 1$
$3 + b = 3 \Rightarrow b = 0$ ; $1 + 0 + c = - 1 \Rightarrow c = - 2$
$T_{n} = n^{2} - 2$
(b) $3 + 2 (n - 1) = 2 n + 1$
(c) $2 n + 1 = 81 \Rightarrow n = 40$
Between $T_{40}$ and $T_{41}$

Question 4

9 marks total

problem-solving

Square tiles are laid in a sequence of figures. Figures 1 to 4 use $5; 12; 21; 32$ tiles.

(a) Determine a formula for the number of tiles in figure $n$ . (4)

(b) Which figure uses 621 tiles? (3)

Worked solution

Hidden - try the question first

Solution

(a) First differences $a = 1$ ; second difference 2
$a = 1$
$3 + b = 7 \Rightarrow b = 4$ ; $1 + 4 + c = 5 \Rightarrow c = 0$
$T_{n} = n^{2} + 4 n$
(b) $n^{2} + 4 n - 621 = 0$
$(n - 23) (n + 27) = 0$
Figure 23
(c) $n^{2} + 4 n - 100 = 0 \Rightarrow n = - 2 \pm 104$
$n \approx 8, 2$ is not a natural number, so no figure has 100 tiles

FAQs

How do I know if a pattern is quadratic?

Find the first differences. If they are not constant but the second differences are, the pattern is quadratic.

Are quadratic patterns examined in Grade 12 or only Grade 11?

Both. Grade 12 papers include them, often linked to arithmetic sequences through the first differences.

What if my value of n is not a whole number?

Then the value is not a term in the pattern. Term numbers must be natural numbers.

Finding the general term of a quadratic pattern

First differences as a linear pattern

Key terms

Common mistakes

Practice questions

FAQs

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