Quadratic Patterns in Context

Q: How do I find the largest term in a pattern?

If $a < 0$, use $n = -\frac{b}{2a}$ and round to the nearest whole number if needed, then find $T_n$.

Q: Does a quadratic pattern always overtake a linear one?

Only when $a > 0$. Then its growth increases every step, while a linear pattern grows by the same amount each time. With $a < 0$ the quadratic pattern eventually decreases, so it does not overtake an increasing linear pattern.

Pattern questions often come from a real situation: tiles, matchsticks, handshakes or savings. You build the pattern from the context, find its formula, and use it to answer questions about the situation.

Patterns from diagrams and arrangements

Count the items in the first three or four diagrams.
Find the first and second differences.
Find $T_{n}$ .
Check with a diagram you did not use.

Example: handshakes. Every person in a group shakes hands with every other person once. For 1, 2, 3, 4, 5 people: $0; 1; 3; 6; 10$ handshakes.

Second difference 1, so $a = \frac{1}{2}$ , $b = - \frac{1}{2}$ , $c = 0$ :

$T_{n} = \frac{1}{2} n^{2} - \frac{1}{2} n = \frac{n ( n - 1 )}{2}$

With 20 people, there are $\frac{20 \times 19}{2} = 190$ handshakes.

Maximum or minimum term

If $a < 0$ , the pattern increases to a maximum and then decreases. If $a > 0$ , it may decrease to a minimum first.

Find the term number of the turning point with $n = - \frac{b}{2 a}$ . If it is not a whole number, check the two nearest natural numbers.

Example. $16; 25; 32; 37; \dots$ has $T_{n} = - n^{2} + 12 n + 5$ .

$n = - \frac{12}{2 ( - 1 )} = 6$ , so the maximum term is $T_{6} = - 36 + 72 + 5 = 41$ .

Comparing linear and quadratic patterns

A linear pattern grows by the same amount each time. A quadratic pattern's growth itself grows. Eventually a quadratic pattern with $a > 0$ overtakes any linear pattern.

Example. Two savings plans:

Plan A: R100 every week. Total after $n$ weeks: $100 n$ .
Plan B: R10 in week 1, R20 in week 2, R30 in week 3, and so on. Totals: $10; 30; 60; 100; \dots$ , so $5 n^{2} + 5 n$ .

Plan B's total is larger when $5 n^{2} + 5 n > 100 n$ , that is when $n > 19$ . From week 20, Plan B is ahead.

Key terms

Context: The real situation a pattern describes.
Maximum term: The largest value a pattern reaches.
Linear pattern: A pattern with a constant first difference.

Common mistakes

Counting the first diagram wrongly, which spoils the whole formula.
Giving the term number instead of the term value for a maximum.
Not checking whether the answer makes sense in context (for example, a fraction of a person).
Comparing patterns at one value of $n$ only.

Practice questions

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

Question 1

4 marks total

routine

In a group of $n$ people, everyone shakes hands with everyone else once. For $n = 1, 2, 3, 4, 5$ the numbers of handshakes are $0; 1; 3; 6; 10$ .

(a) Determine a formula for the number of handshakes. (3)

(b) How many handshakes are there in a group of 20 people? (1)

Worked solution

Hidden - try the question first

Solution

(a) Second difference 1, so $a = \frac{1}{2}$
$3 a + b = 1 \Rightarrow b = - \frac{1}{2}$ ; $a + b + c = 0 \Rightarrow c = 0$
$T_{n} = \frac{1}{2} n^{2} - \frac{1}{2} n$
(b) $T_{20} = 190$

Question 2

5 marks total

routine

Matchsticks are used to make figures. Figures 1 to 4 use $4; 10; 18; 28$ matchsticks.

(a) Determine a formula for the number of matchsticks in figure $n$ . (3)

(b) How many matchsticks are in figure 15? (2)

Worked solution

Hidden - try the question first

Solution

(a) First differences $a = \frac{1}{2}$ ; second difference 2, so $a = 1$
$3 + b = 6 \Rightarrow b = 3$ ; $1 + 3 + c = 4 \Rightarrow c = 0$
$T_{n} = n^{2} + 3 n$
(b) $T_{15} = 225 + 45$
$= 270$

Question 3

6 marks total

complex

Given the quadratic pattern $16; 25; 32; 37; \dots$

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

(b) Determine the maximum term of the pattern. (3)

Worked solution

Hidden - try the question first

Solution

(a) Second difference $a = \frac{1}{2}$ , so $a = - 1$
$- 3 + b = 9 \Rightarrow b = 12$ ; $- 1 + 12 + c = 16 \Rightarrow c = 5$
$T_{n} = - n^{2} + 12 n + 5$
(b) $n = - \frac{12}{2 ( - 1 )} = 6$
$T_{6} = - 36 + 72 + 5$
$= 41$

Question 4

8 marks total

problem-solving

Nomsa compares two savings plans.

Plan A: R100 every week.
Plan B: R10 in week 1, R20 in week 2, R30 in week 3, and so on.

(a) Calculate the total saved after 4 weeks on each plan. (2)

(b) Show that the total saved on Plan B after $n$ weeks is $5 n^{2} + 5 n$ . (3)

Worked solution

Hidden - try the question first

Solution

(a) Plan A: R400
Plan B: $10 + 20 + 30 + 40 =$ R100
(b) Totals $3 a + b = 1 \Rightarrow b = - \frac{1}{2}$ : first differences $20; 30; 40$ ; second difference 10
$a = 5$ ; $15 + b = 20 \Rightarrow b = 5$
$5 + 5 + c = 10 \Rightarrow c = 0$ , so $T_{n} = 5 n^{2} + 5 n$
(c) $5 n^{2} + 5 n > 100 n \Rightarrow 5 n (n - 19) > 0$
$n > 19$
From week 20

FAQs

How do I turn a diagram into a pattern?

Count carefully for the first few diagrams, write the numbers in order, and find the differences.

How do I find the largest term in a pattern?

If $a < 0$ , use $n = - \frac{b}{2 a}$ and round to the nearest whole number if needed, then find $T_{n}$ .

Does a quadratic pattern always overtake a linear one?

Only when $a > 0$ . Then its growth increases every step, while a linear pattern grows by the same amount each time. With $a < 0$ the quadratic pattern eventually decreases, so it does not overtake an increasing linear pattern.

Patterns from diagrams and arrangements

Maximum or minimum term

Comparing linear and quadratic patterns

Key terms

Common mistakes

Practice questions

FAQs

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