Example – Finding the nth term of a quadratic sequence

Here is a sequence of patterns.

Pattern 1 Pattern 2 Pattern 3

a Draw pattern 4.

b The number of squares in the patterns form a sequence.

What are the fi rst fi ve terms of the sequence?

c Find the

n

th term of this sequence.

Explain how the patterns help you to fi nd the

n

th term of the sequence.

Solution

a

Pattern 4

b From the diagrams, the fi rst four terms are 5, 8, 13, 20.

Sequence 5 8 13 20 ...

1st difference +3 +5 +7

2nd difference +2 +2

The next fi rst difference will be 9, so the fi fth term is 29.

c The fi rst differences are not the same so the sequence is not linear.

The second differences are the same so this is a quadratic sequence.

This sequence 5 8 13 20 29

+4

1 4 9 16 25

Square numbers

+4 +4 +4 +4

The

n

th term is therefore

n

^{2 + 4.}

Look at the pattern.

• The fi rst pattern is a 1 × 1 square plus four single squares.

• The second pattern is a 2 × 2 square plus four single squares.

• The third pattern is a 3 × 3 square plus four single squares.

So the

n

th pattern will be an

n

^×

n

square plus four single squares, giving

n

² + 4 squares in total.

Start with the square numbers.

To get the required sequence, add 4.

Unit 5 Quadratic sequences Band g

Practising skills

1 Write down the fi rst fi ve terms for these sequences.

a

n

th term = 5

n

²

b

n

^{th term =}

n

^{2 + 5}

c

n

th term = 2

n

² – 1 d

n

th term = 3

n

^{2 + 4}

e

n

th term = 10

n

² – 5

2 Match each sequence with its position-to-term formula.

a 9, 8, 7, 6, 5, … b 90, 80, 70, 60, 50, … c 10, 40, 90, 160, 250, … d 11, 14, 19, 26, 35, …

3 Find the missing term in each of these sequences and then write down the position-to-term formula.

a 1, 4, 9, , 25 b 2, 5, 10, , 26 c 2, 8, , 32, 50 d 5, 11, 21, , 53

4 a Copy and complete this table for the sequence with

n

^{th term}

n

^{2 + 3.}

(You may fi nd compiling a spreadsheet useful for this.) Term

n

² + 3 1st difference 2nd difference

1 4

2 7 3

3 12 5 2

4 5

b Repeat part a for i

n

^{2 – 1}

iv 5

n

²

ii

n

^{2 + 2}

v 3

n

^{2 + 5}

iii 2

n

^{2 – 3}

c Do all quadratic sequences have the same second difference?

5 Match each sequence with its position-to-term formula.

i 2

n

^{2 + 2 ii}

n

^{2 + 2 iii 4}

n

^{2 + 6 iv 5}

n

^{2 + 5 v 100 – 2}

n

²

a 10, 22, 42, 70, 106, … b 98, 92, 82, 68, 50, … c 4, 10, 20, 34, 52, … d 3, 6, 11, 18, 27, … e 10, 25, 50, 85, 130, …

i

n

² + 10 ii 10 –

n

iii 100 – 10

n

iv 10

n

²

6 a Which of these sequences are quadratic?

i 5, 8, 13, 20, 29, … ii 10, 20, 30, 40, 50, … iii 3, 12, 27, 48, 75, … iv 2, 16, 54, 128, 250, … v 5, 11, 21, 35, 53, …

b For the quadratic sequences, write down the position-to-term formula.

Developing fl uency

1 Look at this table of sequences.

a Write down the next term in each sequence.

b Write down the rule for the

n

th term of sequence A and B.

c i How is sequence C related to sequence B?

ii Write down the

n

th term of sequence C.

d i How is sequence D related to sequences A and B?

ii Write down the

n

th term of sequence D.

e i Which two sequences are used to make sequence E?

ii Write down the

n

th term of sequence E.

2 Katrina is making a sequence of patterns from square tiles.

a Draw the next pattern in the sequence.

b Copy and complete the table.

Pattern 1 2 3 4 5

Number of tiles

c Which pattern uses 100 tiles?

d How many tiles are there in the

n

th pattern?

Katrina removes some tiles from each pattern to make the following sequence of patterns.

e i How many tiles does Katrina remove from the 50th pattern?

ii How many tiles are left in the 50th pattern?

iii How many tiles are left in the

n

th pattern?

1 2 3 4 5 …

A 2 4 6 8 10 …

B 1 4 9 16 25 … C 0 3 8 15 24 … D 3 8 15 24 35 … E 1 3 14 23 34 …

Pattern 1 Pattern 2 Pattern 3

Pattern 1 Pattern 2 Pattern 3 Pattern 4

Unit 5 Quadratic sequences Band g

3 Here is a pattern made from triangular tiles.

a How many tiles would be needed for Pattern number 8?

b How many tiles are needed for Patter number

n

^?

c Which pattern has 100 tiles?

d Explain why the difference between 2 adjacent pattern numbers is always an odd number.

4 Ben is stacking tins of baked beans.

a How many tins are in stack 5?

The 20th stack needs 210 tins.

b How many tins are needed for the 21st stack?

Ben has 120 tins to stack.

c How many tins should he place in the bottom row?

The

n

th stack has 1

2

n

(

n

+ 1) tins.

d How many tins are in the 100th stack?

Comfort has 169 tins to stack.

e i Can she make one stack out of these tins?

Explain your answer fully.

ii Comfort uses all of the 169 tins to make 2 stacks.

How many tins are in each stack?

5 Look at this Rubik’s cube.

It is made up of small cubes around a central mechanism.

a Write down how many cubes have

i 1 sticker ii 2 stickers iii 3 stickers.

b How many cubes have at least one sticker on them?

c You can get different-sized Rubik’s cubes.

The smallest is a 2 by 2 by 2 cube. You can also get larger cubes like a 5 by 5 by 5 cube.

Copy and complete this table for different-sized cubes.

Cube size 2 3 4 5 10

n

1 sticker 2 stickers 3 stickers Total number of stickered cubes

Pattern number 1 Pattern number 2 Pattern number 3

Reasoning

Stack 1 Stack 2 Stack 3

Reasoning

Problem solving

1 Here are the fi rst 3 shapes in a rectangular pattern made from dots.

a How many dots are there in pattern number 6?

b Find the number of dots in the

n

th pattern.

c Find an expression, in terms of

n

, for the difference in the number of dots between the

n

th pattern and the (

n

+ 1) pattern.

d Between which two consecutive patterns is the difference in the number of dots 102?

2 Jake draws a circle. He marks points around the circumference of the circle, joining each point to every other point.

Points Number of lines

at each point Total number of lines

2 1 1

3 2 3

4 3 6

5 4

6 5

7 7

8 8

n

a Copy and complete this table for the number of points around the circle.

b Mary made a circle pattern by marking points at every 10 º from the centre of the circle.

How many lines did Mary draw?

3 The

n

th term of a quadratic sequence is

n

² + 8.

The

n

th term of a different quadratic sequence is 49 –

n

^2.

Which numbers are in both sequences?

4 The diagrams show the numbers of diagonals in some regular polygons.

a Find the number of diagonals in a regular decagon.

b Find an expression, in terms of

n,

for the number of diagonals at each vertex of an

n

-sided polygon.

c Hence, or otherwise, fi nd the number of diagonals in an

n

-sided regular polygon.

d An

n

-sided polygon has over 100 diagonals. What is the smallest possible value of

n

^?

Exam-styleExam-styleExam-styleExam-style

Unit 5 Quadratic sequences Band g

Reviewing skills

1 Find the missing term in each of these sequences and then write down the position-to-term formula.

a 20, 30, 40, 50, 60, , … b 3, 12, 27, , 75, … c 99, 96, 99, , 75, … d 12, 45, 100, , 276, …

2 Copy and complete this table for the sequence with

n

th term

n

² + 5.

(You may fi nd compiling a spreadsheet useful for this.) Term

n

² + 5 1st difference 2nd difference

1 6

2 9 3

3 14 5 2

4 5

3 Here are some patterns made from square tiles.

Pattern 1 Pattern 2 Pattern 3

a Draw pattern number 4.

b Copy and complete the table.

Pattern number 1 2 3 4 5

Number of green squares Number of blue squares

Number of red squares Total number of squares, S

c Write down how many of these are in the 10th pattern.

i green squares ii blue squares iii red squares iv Work out the total number of squares in the 10th pattern.

d Write down how many of these are in the

n

th pattern.

i green squares ii blue squares iii red squares

e Write down a formula for the total number of squares, S, in the

n

th pattern.

Write your formula in two different ways.

2

Outside the Maths classroom

In document Mathematics for Edexcel GCSE - Foundation 2-Higher 1 (2015) (Page 113-119)