Quadratic Number Patterns | IEB Grade 11 Maths

Q: What is the formula for a quadratic pattern?

$T_n = an^2 + bn + c$, where $2a$ is the second difference, $3a + b$ is the first first difference and $a + b + c$ is the first term.

Quadratic number patterns are new in Grade 11. The second differences are constant, which lets you find a formula for any term. Patterns carry about 25 marks in Paper 1, and the same skills continue into Grade 12.

First and second differences

First differences: subtract each term from the next.
Second differences: subtract each first difference from the next.

If the first differences are constant, the pattern is linear. If the second differences are constant, it is quadratic.

Example. $2; 5; 10; 17; \dots$

First differences: $3; 5; 7$ . Second differences: $2; 2$ . The pattern is quadratic, and the next terms are $17 + 9 = 26$ and $26 + 11 = 37$ .

The first differences of a quadratic pattern form a linear pattern.

Finding Tn = an² + bn + c

Match the pattern with these three equations:

$2 a$ = second difference
$3 a + b = T_{2} - T_{1}$ (the first of the first differences)
$a + b + c = T_{1}$

Example. $1; 6; 13; 22; \dots$

First differences $5; 7; 9$ , second difference 2.

$a = 13 + b = 5 \Rightarrow b = 21 + 2 + c = 1 \Rightarrow c = - 2$

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

Check: $T_{3} = 9 + 6 - 2 = 13$ ✓

Finding a term or a term number

A term: substitute $n$ . For the pattern above, $T_{20} = 400 + 40 - 2 = 438$ .

A term number: set $T_{n}$ equal to the value and solve. The answer must be a natural number.

Example. Which term is 397?

$n^{2} + 2 n - 2 = 397 \Rightarrow n^{2} + 2 n - 399 = 0 \Rightarrow (n + 21) (n - 19) = 0 \Rightarrow n = 19$

Is a number in the pattern? For 120: $n^{2} + 2 n - 122 = 0$ gives $n = - 1 \pm 123$ , which is not a natural number. So 120 is not a term.

Key terms

First difference: The difference between consecutive terms.
Second difference: The difference between consecutive first differences.
General term: A formula $T_{n}$ for the $n$ th term.
Term number: The position $n$ of a term; a natural number.

Common mistakes

Setting $a$ equal to the second difference instead of half of it.
Using the second first difference in $3 a + b$ instead of the first one.
Keeping a negative or fractional $n$ .
Not checking the formula with a known term.

Practice questions

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

Question 1

3 marks total

knowledge

Write down the next two terms of the pattern $2; 5; 10; 17; \dots$

Worked solution

Hidden - try the question first

Solution

First differences $3; 5; 7$ ; second difference 2
$T_{5} = 17 + 9 = 26$
$T_{6} = 26 + 11 = 37$

Question 2

5 marks total

routine

For the pattern $1; 6; 13; 22; \dots$ , determine $T_{n}$ and $T_{20}$ .

Worked solution

Hidden - try the question first

Solution

$2 a = 2 \Rightarrow a = 1$
$3 a + b = 5 \Rightarrow b = 2$
$a + b + c = 1 \Rightarrow c = - 2$
$T_{n} = n^{2} + 2 n - 2$
$T_{20} = 438$

Question 3

6 marks total

complex

For the pattern $1; 6; 13; 22; \dots$ with $T_{n} = n^{2} + 2 n - 2$ :

(a) Which term is equal to 397? (3)

(b) Is 120 a term of the pattern? Explain. (3)

Worked solution

Hidden - try the question first

Solution

(a) $3; 5; 7$
$(n + 21) (n - 19) = 0$
$n = 19$ , so $T_{19} = 397$
(b) $n^{2} + 2 n - 122 = 0$
$n = - 1 \pm 123$
This is not a natural number, so 120 is not a term

Question 4

8 marks total

problem-solving

$5; p; 21; q; 45$ is a quadratic pattern with a constant second difference of 2.

(a) Determine $p$ and $q$ . (4)

(b) Determine $T_{n}$ . (2)

Worked solution

Hidden - try the question first

Solution

(a) Let the first difference be $3; 5; 7$ ; the next is $d + 2$
$T_{5} = 17 + 9 = 26$
$p = 12$
$q = 21 + 11 = 32$ (check: $32 + 13 = 45$ )
(b) $a = 1$ ; $3 + b = 7 \Rightarrow b = 4$ ; $c = 0$
$T_{n} = n^{2} + 4 n$
(c) $T_{50} = 2500 + 200$
$= 2700$

FAQs

How do I know if a pattern is quadratic?

Its first differences change, but its second differences are constant.

What is the formula for a quadratic pattern?

$T_{n} = a n^{2} + bn + c$ , where $2 a$ is the second difference, $3 a + b$ is the first first difference and $a + b + c$ is the first term.

Are quadratic patterns examined in Grade 12?

Yes, often linked to arithmetic sequences through the first differences.

Quadratic number patterns: the general term

First and second differences

Finding Tn = an² + bn + c

Finding a term or a term number

Key terms

Common mistakes

Practice questions

FAQs

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