GCSE Maths · Higher · Chapter 4: Number and sequences · Lesson 3

Quadratic sequences

Grades 6–8About 50 minutesNo calculator40 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Find the 18th term of the sequence with $n$ th term $4 n - 3$ .
[1 mark]
Answer
A2
Last lesson
Each term is the sum of the two before. Find the next term: $2, 2, 4, 6, 10, \dots$
[1 mark]
Answer
A3
Key Stage 3
Write $32 : 72$ in its simplest form.
[1 mark]
Answer
A4
Key Stage 3
How many minutes are in $0.6$ of an hour?
[1 mark]
Answer

Grades 6–7

Your teacher works through each example with you. Then try the one beside it.

Example 1

Find the $n$ th term of the sequence $6$ , $13$ , $24$ , $39$ , $58$ , ...

Hint Halve the second difference to get the number in front of

n^{2}

. Subtract that many

n^{2}

from each term, then find the linear rule for what is left.

B1
Find the $n$ th term of the sequence $4$ , $9$ , $20$ , $37$ , $60$ , ...
[3 marks]
Answer

Example 2

The $n$ th term of a sequence is $n^{2} + 2 n - 5$ . Which term is equal to $115$ ?

Hint Set the

n

th term equal to

115

, rearrange to equal

0

and factorise. A position cannot be negative.

B2
The $n$ th term of a sequence is $n^{2} + 5 n - 2$ . Which term is equal to $124$ ?
[3 marks]
Answer

Grades 6–7

C1
Show that $5$ , $9$ , $15$ , $23$ , $33$ , ... is a quadratic sequence, and work out the next term.
[2 marks]
Answer
C2
Find the $n$ th term of the sequence $14$ , $17$ , $22$ , $29$ , ...
[2 marks]
Answer

C3
Find the $n$ th term of the sequence $8$ , $32$ , $72$ , $128$ , ...
[2 marks]
Answer
C4
Is $200$ a term of the sequence with $n$ th term $2 n^{2} - 2$ ? You must show your working.
[2 marks]
C5
The $n$ th term of a sequence is $n^{2} + 8 n$ . Find the position of the first term that is greater than $300$ .
[2 marks]
Answer
C6
The first three terms of a quadratic sequence are $5$ , $13$ , $27$ . Work out the $6$ th term. You must show your working.
[3 marks]
Answer

Grades 7–8

D1
Here are the first five terms of a sequence: $15$ , $18$ , $23$ , $30$ , $39$ . Leo says, "The $n$ th term is $3 n + 12$ , because the first difference is $3$ ."
Explain why Leo is wrong. Find the correct $n$ th term.
[3 marks]

D2
The diagram shows the first three patterns in a sequence. Pattern $n$ is a square of $n \times n$ white tiles with a border of grey tiles all the way round it.
(a)
Find an expression, in terms of $n$ , for the total number of tiles in pattern $n$ .
[2 marks]
Answer
(b)
Which is the first pattern with more white tiles than grey tiles? You must show your working.
[3 marks]
Answer
(c)
One pattern uses $400$ tiles in total. How many of them are grey?
[2 marks]
Answer
Total for D2: 7 marks
D3
A quadratic sequence has $n$ th term $2 n^{2} + bn + c$ . Its first term is $7$ and its second term is $16$ . Work out the $5$ th term of the sequence. You must show your working.
[4 marks]
Answer

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can show a sequence is quadratic and find its $n$ th term.
I can use an $n$ th term to find a term, a position, or show a number is not a term.
I can find a quadratic rule from a pattern or from given terms.

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