nth term of a linear sequence Quiz (Grade 4)

Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8/9

Topic:

Score: 0 / 10

nth term of a linear sequence — Notes & Worked Examples Show / Hide

Explanation:

To find the nth term of a linear (arithmetic) sequence, use the "DINO" method: find the common Difference d between terms and write dn, then find the 0th term (work backwards one step before the first term) and write it on the end.

To test whether a number is a term in the sequence, set the nth term equal to that number and solve for n. If n is a positive whole number (integer), then the number is in the sequence; if n is a fraction or decimal, it is not.

Example 1: Find the nth term and the 50th term of the sequence: 7, 11, 15, 19, 23, …

Solution:

Common difference = +4 → starts with 4n

0th term (before 7) = 7 − 4 = +3

nth term = 4n + 3

50th term (n = 50): 4(50) + 3 = 203

Answer: 4n + 3; 50th term = 203

Example 2: A sequence has nth term 6n − 5. Is 148 a term in this sequence? You must show your working.

Solution:

Set 6n − 5 = 148

6n = 153 → n = 153 ÷ 6 = 25.5

Since 25.5 is not a whole number (integer),

Answer: No, 148 is not in the sequence.

One Page Guide (PDF) nth term of a linear sequence Worksheets

Question 1

[1 mark]

Find the next three terms of the sequence
4, 7, 10, 13, 16, \hspace{0.4cm}, \hspace{0.4cm}, \hspace{0.4cm}
9, 17, 25, 33, 41, \hspace{0.4cm}, \hspace{0.4cm}, \hspace{0.4cm}

Worked Solution & Answer

Add 3
Add 8 B1

Answer: 19, 22, 25 and 49, 57, 65

Question 2

[1 mark]

Find the next two terms
4, 8, 12, 16, \hspace{0.5cm}, \hspace{0.5cm}

Worked Solution & Answer

Add 4 each time B1

Answer: 20, 24

Question 3

[2 marks]

Here is a sequence of patterns made from grey square tiles.

(a) On the grid below, draw Pattern number 5 [1 mark]

(b) Complete the table. [1 mark]

Worked Solution & Answer

(a)

L-shape with 5 squares along top and 5 squares down left (9 squares total)

Answer (a): L-shape with 5 squares along top and 5 squares down left (9 squares total)

(b)

Number of squares increases by 2 each time: 1, 3, 5, 7, 9, 11 B1

Answer (b): 9, 11

Question Paper (PDF) • Mark Scheme (PDF)

Question 4 AQA Jun 2025 3F Q1

[3 marks]

Here are the first three patterns in a sequence.

(a) Draw Pattern 4 on the grid. [1 mark]

(b) How many squares in Pattern 6 would have a cross (X) in them? [2 marks]

Answer

Worked Solution & Answer

(a)

Number of crosses follows the sequence: 5, 6, 7, 8, ..
Pattern 4 has 8 crosses (6 on top row, 2 on bottom row corners) B1

Answer (a): Pattern 4 drawn: top row of 6 crosses, bottom row with cross at left, cross at right, 4 shaded squares in between (total 8 crosses).

(b)

Pattern 1: 5 crosses, Pattern 2: 6 crosses, Pattern 3: 7 crosses, Pattern n: n + 4 crosses M1
For Pattern 6: 6 + 4 = 10 crosses A1

Answer (b): Answer 10

Question Paper (PDF) • Mark Scheme (PDF)

Question 5 OCR Jun 2024 1F Q4

[3 marks]

Here are the first three dot patterns in a sequence.

(a) Draw Pattern 4 in the sequence. [1 mark]

(b) Without drawing, work out how many dots are in Pattern 10 of the sequence.
Explain how you worked out your answer. [2 marks]

Worked Solution & Answer

(a)

Pattern 4 has 4 dots on each side of the centre dot horizontally and 4 dots above it (13 dots). B1

Answer (a): Pattern 4 drawn with 9 horizontal dots and 4 vertical dots above the centre (13 dots in total)

(b)

Pattern 1 = 4, Pattern 2 = 7, Pattern 3 = 10 M1
Rule is 3n + 1, so for Pattern 10: 3 × 10 + 1 = 31. C1

Answer (b): 31, because 3 dots are added each time (3 × 10 + 1 = 31).

Question Paper (PDF) • Mark Scheme (PDF)

Question 6 OCR Jun 2022 3F Q3

[3 marks]

Here are the first four dot patterns in a sequence.

(a) Draw Pattern 5 in the sequence. [1 mark]

(b) Without drawing, work out how many dots are in Pattern 8 of the sequence.
Explain how you worked out your answer. [2 marks]

Worked Solution & Answer

(a)

Draw 5 vertical dots on the left and 4 vertical dots on the right (9 dots). B1

Answer (a): Pattern 5 drawn with 5 dots in left column and 4 dots in right column (9 dots total)

(b)

The numbers of dots are 1, 3, 5, 7, 9, ... adding 2 each time M1
Pattern 8 has 2 × 8 − 1 = 15 dots. C1

Answer (b): 15 dots, because 2 dots are added each time (or 2 × 8 − 1 = 15).

Question Paper (PDF) • Mark Scheme (PDF)

Question 7

[4 marks]

Here is a sequence of patterns made with counters.

(a) Find an expression, in terms of n, for the number of counters in pattern number n. [2 marks]

(b) Can Bayo make a pattern in this sequence using all 90 of his counters?
You must show how you get your answer. [2 marks]

Worked Solution & Answer

(a)

Counts: 4, 7, 10 → common difference 3 → 3n + 1 M1 A1

Answer (a): 3n + 1

(b)

3n + 1 = 90 M1
⇒ 3n = 89
⇒ n = 29.67 (not an integer) A1

Answer (b): No, because 3n + 1 = 90 gives 3n = 89, and 89 ÷ 3 = 29.67 is not a whole number (pattern 29 uses 88 counters and pattern 30 uses 91)

Question Paper (PDF) • Mark Scheme (PDF)

Question 8 OCR Nov 2024 2F Q8

[4 marks]

(a) (i) A sequence is generated using the rule:
\textsf{multiply the previous term by } 3\textsf{, then subtract } 1.
The **2nd** term of the sequence is 20.
Find the **3rd** term of the sequence.
(ii) Find the **1st** term of the sequence.
(b) Here are the first four terms of a different sequence.
5 10 15 20
Find the nth term of the sequence.

(a) A sequence is generated using the rule:
\textsf{multiply the previous term by } 3\textsf{, then subtract } 1.
The **2nd** term of the sequence is 20.
Find the **3rd** term of the sequence. [1 mark]

(b) Find the **1st** term of the sequence. [2 marks]

(b) Here are the first four terms of a different sequence.
5 10 15 20
Find the nth term of the sequence. [1 mark]

Worked Solution & Answer

(a)

20 × 3 − 1 = 59. B1

Answer (a): 59

(b)

(20 + 1) ÷ 3 = 21 ÷ 3 = 7. M1 A1

Answer (b): 7

(b)

Multiples of 5 ⇒ nth term is 5n. B1

Answer (b): 5n n

Question Paper (PDF) • Mark Scheme (PDF)

Question 9 Eduqas Jun 2025 2F Q4

[4 marks]

Work out
(a) Write the next term in the sequence below.
4, 28, 52, 76, ...
(b) Draw Diagram 1 in the sequence below.
(c) The sequence of numbers shown below is linear.
31, ..., ..., 16, 11
Two of the numbers are missing.
Fill in the missing numbers in the sequence.
Write down the rule for finding the next term in the sequence.

(a) Write the next term in the sequence below.
4, 28, 52, 76, ... [1 mark]

(b) Draw Diagram 1 in the sequence below. [1 mark]

(c) The sequence of numbers shown below is linear.
31, ..., ..., 16, 11
Two of the numbers are missing.
Fill in the missing numbers in the sequence.
Write down the rule for finding the next term in the sequence. [2 marks]

Worked Solution & Answer

(a)

Common difference is +24:
76 + 24 = 100 B1

Answer (a): 100

(b)

Diagram numbers follow triangular patterns: Diagram 4 has base 5, Diagram 3 has base 4, Diagram 2 has base 3. Therefore, Diagram 1 has base 2 with 2 dots in column 1 and 1 dot in column 2 (total 3 dots). B1

Answer (b): A column of 2 dots and a column of 1 dot (3 dots in total forming a triangle of base 2 and height 2)

(c)

Common difference between 16 and 11 is 11 − 16 = −5. M1
Missing numbers: 31 − 5 = 26, 26 − 5 = 21
Check: 21 − 5 = 16 ✓
Rule: Subtract 5. A1

Answer (c): Missing numbers: 26, 21; Rule: Subtract 5 (or −5)

Question Paper (PDF) • Mark Scheme (PDF)

Question 10 IGCSE Jan 2019 1FR Q9

[5 marks]

Here is a sequence of patterns made from white counters and black counters.

(a) In the space below, complete Pattern number 4 [1 mark]

(b) Find the total number of counters in Pattern number 6 [1 mark]

(d) Work out the total number of counters in Pattern number 50 [2 marks]

Worked Solution & Answer

(a)

Pattern 4 drawn with 9 counters

Answer (a): Pattern 4 drawn with 9 counters

(b)

Answer (b): 13

(c)

Answer (c): 15

(d)

Pattern n has 2n + 1 counters M1
2 × 50 + 1 = 101 A1

Answer (d): 101

Question Paper (PDF) • Mark Scheme (PDF)

Score: 0 / 10 (0 / 30 marks)

nth term of a linear sequence Worksheets Next: Expanding and factorising →

Grade 4 Quizzes

24 Topics

4A - Compound interest	4B - Indices (index laws)
4C - Prime factors HCF and LCM	4D - Real life graphs
4E - Inequalities on a number line	4F - Solving inequalities
4G - Forming and solving equations	4H - nth term of a linear sequence
4J - Expanding and factorising	4K - Pythagoras theorem
4L - Angles in parallel lines	4M - Angles in polygons
4N - Surface area	4P - Volume of a cuboid
4Q - Volume of a prism	4R - Bearings
4S - Cylinders	4T - Plans and elevations
4U - Scatter graphs	4V - Probability
4W - Averages from tables	4X - Reverse mean
4Y - Area of a trapezium	4Z - Loci and construction