Year 7 Maths: Exponential Growth and Sequences

Q: What is the 5th term in the geometric sequence: 3, 6, 12, 24, ...?

Correct Answer: B) 48. Explanation: Each term is multiplied by 2 (ratio = 2). 5th term = 24 × 2 = 48. Mark Scheme: 1 mark for 48.

Q: Which of these sequences represents EXPONENTIAL growth?

Correct Answer: B) 3, 9, 27, 81, 243.... Explanation: Sequence B multiplies by 3 at each step (3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81...), representing exponential growth. Mark Scheme: 1 mark for selecting sequence B.

Q: A bank deposit of £100 doubles every 10 years. How much money is in the account after 30 years?

Correct Answer: C) £800. Explanation: 30 years = 3 doubling periods. 100 → £200 (10 yrs) → £400 (20 yrs) → £800 (30 yrs). Mark Scheme: 1 mark for £800.

Key Learning Objectives

By the end of this lesson and revision module, Year 7 students will be able to:

Understand the difference between arithmetic (linear) sequences and geometric (exponential) sequences.
Calculate terms in geometric sequences involving repeated multiplication by a constant ratio.
Model real-life exponential phenomena such as population growth, compound interest, and doubling patterns.

Topic Summary & Essential Notes

In Year 7 Mathematics, students compare Linear (Arithmetic) Sequences and Exponential (Geometric) Sequences. In a linear sequence, you add or subtract the same common difference each time (e.g. 3, 6, 9, 12... adding 3). In an exponential sequence, you multiply by a constant multiplier or ratio each time (e.g. 2, 4, 8, 16, 32... doubling each step). While linear growth creates a straight diagonal line on a graph, exponential growth starts slowly and then curves upward sharply, growing at an astonishing rate. The famous 'Wheat and Chessboard' problem illustrates exponential power: placing 1 grain on square 1, 2 on square 2, 4 on square 3 leads to over 18 quintillion grains by square 64!

Key Rules, Formulas & Definitions

Linear Sequence Formula

Term n = an + b (constant addition/subtraction, straight line graph)

Geometric / Exponential Sequence

Term n = a × rⁿ⁻¹ (constant multiplication by ratio r, exponential curve)

Doubling Formula

After n doublings, Value = Initial Amount × 2ⁿ

2 Step-by-Step Worked Examples

Study these model solutions to understand how examiners award method and accuracy marks:

Worked Example 1

Worked Example 1: Calculating Terms in a Doubling Exponential Sequence

Question: A pond lily pad covers 1 m² on Day 1. Each day, the lily pad doubles in area. Calculate its area on Day 6.

Step-by-Step Solution:

Step 1: List the daily progression. Day 1 = 1 m², Day 2 = 2 m², Day 3 = 4 m², Day 4 = 8 m², Day 5 = 16 m², Day 6 = 32 m².
Step 2: Formula check. Area = 1 × 2⁵ = 32 m².
Step 3: Verify. Doubling 5 times: 1 → 2 → 4 → 8 → 16 → 32 m².

Final Answer: 32 m²

Senior Examiner Insight: Notice that to find the 6th day starting from Day 1, you multiply by 2 FIVE times (2⁵), not 6 times.

Worked Example 2

Worked Example 2: Distinguishing Linear from Exponential Sequences

Question: Sequence A: 5, 10, 15, 20, 25... | Sequence B: 5, 10, 20, 40, 80... Identify which is linear and which is exponential, and find the 6th term of each.

Step-by-Step Solution:

Step 1: Analyze Sequence A. Common difference is +5 (Linear). 6th term = 25 + 5 = 30.
Step 2: Analyze Sequence B. Common ratio is ×2 (Exponential). 6th term = 80 × 2 = 160.

Final Answer: Sequence A is Linear (6th term = 30); Sequence B is Exponential (6th term = 160).

Senior Examiner Insight: Always test both the difference (+/-) and ratio (×/÷) between consecutive terms.

Common Pitfalls & Examiner Warnings

Avoid these common mistakes frequently identified by examiners in Key Stage 3 assessments:

Common Mistake: Confusing multiplying by 2 with adding 2

In exponential doubling, values multiply by 2 (2, 4, 8, 16...), unlike adding 2 (2, 4, 6, 8...).

Common Mistake: Off-by-one errors with power exponents

Starting at term 1 with value a, the nth term is a × rⁿ⁻¹ (power is n − 1).

Common Mistake: Assuming exponential graphs are straight lines

Exponential graphs form steep J-shaped curves, not straight diagonal lines.

Interactive Self-Check Quiz (3 Questions)

Test your understanding before checking the full worked answer and examiner mark scheme:

Question 1 of 3

What is the 5th term in the geometric sequence: 3, 6, 12, 24, ...?

A) 30

B) 48

C) 36

D) 96

Click to Reveal Answer & Mark Scheme

Correct Answer: B) 48

Detailed Explanation: Each term is multiplied by 2 (ratio = 2). 5th term = 24 × 2 = 48.

Mark Scheme & Scoring Points: 1 mark for 48.

Question 2 of 3

Which of these sequences represents EXPONENTIAL growth?

A) 10, 20, 30, 40, 50...

B) 3, 9, 27, 81, 243...

C) 100, 90, 80, 70...

D) 4, 8, 12, 16, 20...

Click to Reveal Answer & Mark Scheme

Correct Answer: B) 3, 9, 27, 81, 243...

Detailed Explanation: Sequence B multiplies by 3 at each step (3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81...), representing exponential growth.

Mark Scheme & Scoring Points: 1 mark for selecting sequence B.

Question 3 of 3

A bank deposit of £100 doubles every 10 years. How much money is in the account after 30 years?

A) £300

B) £600

C) £800

D) £400

Click to Reveal Answer & Mark Scheme

Correct Answer: C) £800

Detailed Explanation: 30 years = 3 doubling periods. 100 → £200 (10 yrs) → £400 (20 yrs) → £800 (30 yrs).

Mark Scheme & Scoring Points: 1 mark for £800.

Created by Fiaraz Iqbal

Former Headteacher & Senior Science / Maths Examiner

With over 15 years of leadership and classroom experience, Fiaraz creates high-yield revision resources aligned with National Curriculum standards to help Key Stage 3 and GCSE students secure top grades.

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