Edexcel November 2023 1H Question 22

Geometric sequences are a key topic in GCSE maths, particularly when they involve surds. Today, we'll walk through a challenging exam-style question that combines both concepts. This type of question appears on higher-tier papers from all major UK exam boards (AQA, Edexcel, OCR).

Understanding the Question

Let's start by recalling what a geometric sequence is:

> A geometric sequence is one where each term is found by multiplying the previous term by a constant value called the common ratio (usually denoted by r).

If the first term is a, then:

2nd term = ar

3rd term = ar²

• 4th term = ar³

Our given information:

2nd term = 3 + 2√2

• 3rd term = 13 + 9√2

We need to find the common ratio r in the form a + √b where a and b are integers.

Step-by-Step Solution

Step 1: Set up equations using the geometric sequence definition

Since the 2nd term is ar and the 3rd term is ar², we can write:

1. 1) ar = 3 + 2√2
   
2. 2) ar² = 13 + 9√2

Step 2: Find the common ratio by division

The common ratio r is what we multiply by to get from the 2nd term to the 3rd term. Therefore:

r = (3rd term) ÷ (2nd term) = (ar²) ÷ (ar)

This simplifies to:

r = (13 + 9√2) ÷ (3 + 2√2)

Step 3: Simplify the surd expression

We need to simplify ^{13 + 9√2}⁄_{3 + 2√2}.

To do this, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of (3 + 2√2) is (3 - 2√2).

flowchart TD
    A["Start: r = 13+9√2 / 3+2√2"] --> B["Multiply numerator and denominator, by conjugate 3-2√2"]
    B --> C["Expand numerator using FOIL"]
    B --> D["Expand denominator using FOIL"]
    C --> E["Simplify numerator"]
    D --> F["Simplify denominator"]
    E --> G["Divide numerator by denominator"]
    F --> G
    G --> H["Final answer in form a+√b"]

Let's work through this:

r = ^{13 + 9√2}⁄_{3 + 2√2} × ^{3 - 2√2}⁄_{3 - 2√2}

1. Numerator: (13 + 9√2)(3 - 2√2)
   = 13×3 + 13×(-2√2) + 9√2×3 + 9√2×(-2√2)
   = 39 - 26√2 + 27√2 - 18×(√2)²
   = 39 + √2(-26 +
2. 27) - 18×2 [since (√2)² = 2]
   = 39 + √2(1) - 36
   = 3 + √2

Denominator: (3 + 2√2)(3 - 2√2)
This is a difference of two squares: (3)² - (2√2)²
= 9 - (4×2) [since (2√2)² = 4×2 = 8]
= 9 - 8
= 1

Step 4: Complete the division

r = ^{3 + √2}⁄₁ = 3 + √2

Checking Our Answer

Let's verify this makes sense:

If 2nd term = 3 + 2√2 and r = 3 + √2

• Then 3rd term should be (2nd term) × r = (3 + 2√2)(3 + √2)

Check: (3 + 2√2)(3 + √2) = 9 + 3√2 + 6√2 + 2×2 = 9 + 9√2 + 4 = 13 + 9√2 ✓

Perfect! Our answer matches the given 3rd term.

Final Answer

The common ratio is 3 + √2.

In the requested form a + √b:

a = 3

• b = 2

Key Learning Points

Remember the structure: In a geometric sequence with first term a and common ratio r:

• nth term = arⁿ⁻¹

1. Finding r from consecutive terms: r = ^{later term}⁄_{earlier term}
2. Simplifying surd fractions: Always multiply numerator and denominator by the conjugate of the denominator.
3. The conjugate trick: For p + q√b, the conjugate is p - q√b. Multiplying gives p² - q²b (a rational number).

Common Mistakes to Avoid

Forgetting to rationalise: Leaving a surd in the denominator wouldn't give the form a + √b requested.

Incorrect expansion: When expanding brackets with surds, remember that (√2)² = 2, not √4.

• Misidentifying terms: Ensure you're dividing the correct terms - the 3rd term by the 2nd term, not vice versa.

Practice This Technique

Try this similar question:
> The 3rd term of a geometric sequence is 5 + 3√3
> The 4th term is 14 + 8√3
> Find the common ratio in the form a + √b.

(Answer: 2 + √3)

Exam Tips

Show all working: This question specifically asks for it. Even if your final answer is wrong, you can earn method marks.

Check your answer: Multiply the 2nd term by your found ratio to see if you get the 3rd term.

1. Form presentation: Ensure your final answer is in the exact form requested (a + √b with a and b integers).

Geometric sequences with surds combine two important GCSE topics. Mastering this type of question will help you tackle similar problems confidently in your exams. Remember the conjugate method for simplifying surd fractions - it's a powerful tool that works every time!

1. Need more help with geometric sequences or surds? Check out our other GCSE revision guides on these topics.