 # Explore Sequences Involving Surds

In this worksheet, students will find the common ratios in sequences involving surds, apply these to find missing terms and explore sequential relationships. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, OCR, Pearson Edexcel

Curriculum topic:   Algebra

Curriculum subtopic:   Sequences

Difficulty level:   ### QUESTION 1 of 10

If you are aiming for the top grades, you need to be able to tie together two important topics: sequences and surds.

The good news is that surds only appear in one type of sequence: geometric sequences.

What is special about a geometric sequence?

A geometric sequence is one that has a common ratio.

This is where the next term is generated by multiplying the previous term.

e.g. the sequence 2, 4, 8, 16, ... is generated by multiplying by 2.

How to find the common ratio

Great news - this is surprisingly easy!

All we have to do is to divide one term by the previous term.

This will always give the common ratio.

e.g. if we divide the last term in the sequence above (16) by the second to last term (8), we reach the common ratio (2).

Let's look at this in action with some sequences involving surds now.

e.g. Continue the sequence √3, 3, 3√3, 9, ...

Step 1: Find the common ratio.

We're going to pick the third and second terms for this, just because the numbers are easier to work with.

You can pick whichever ones you choose to find the common ratio.

3√3 ÷ 3 = √3

So my common ratio is √3.

Step 2: Continue the sequence.

Now we need to multiply the fourth term by √3 to get the fifth term:

9 × √3 = 9√3

Then I can multiply this fifth term by √3 to get the sixth term:

9√3 × √3 = 9 × 3 = 27

In this activity, you will find the common ratios in sequences involving surds and apply these to find missing terms and explore sequential relationships.

If you see a surd in a sequence, which type of sequence can you assume it is?

What is the next term in this sequence?

4, 4√3, 12, 12√3, ...

Which of the options below is the correct common ratio for this sequence?

1√5, 5, 5√5, ...

5

√5

÷ √5

Match each sequence below to its correct common ratio.

## Column B

2, 2√2, 4, 4√2, ...
√2
9, 3√3, 3, √3, ...
√3
-10√2, -40, -80√2, ...
2√2

What is the common ratio for this sequence?

32√6, 16√6, 8√6, 4√6, ...

Consider the sequence below:

√2, 2, 2√2, ...

What is the ninth term in this sequence?

12, 6√2, 6, 3√2, ...

One of the terms in the sequence below is incorrect:

2√2, 4, 4√2, 8√2, ...

Which one is it?

2√2

4

4√2

8√2

Consider the sequence below:

-6, 12√3, -72, 144√3, ...

The common ratio for this sequence can be written in the form a√b.

Find the value of a and b.

2√2

4

4√2

8√2

True or false?

The sequence 8, 4√2, 4, ... will never become negative.

True

False

• Question 1

If you see a surd in a sequence, which type of sequence can you assume it is?

Geometric
geometric
EDDIE SAYS
At this level, surds will usually appear in only geometric sequences. So if you see surds in sequences, it is pretty safe to assume that you are dealing with a geometric sequence. Whilst it is possible to have linear sequences involving surds, they are incredibly easy to calculate, so they are very unlikely to come up.
• Question 2

What is the next term in this sequence?

4, 4√3, 12, 12√3, ...

36
EDDIE SAYS
Remember to find the common ratio to start. We do this by dividing any term by the previous term: 4√3 ÷ 4 = √3 So our common ratio is √3. Then we can multiply the final term given by this ratio to find the next term in the sequence: 12√3 × √3 = 36 Did you find that answer? Review the Introduction now if not, so that you can revise the method to apply before you move on to the rest of this activity.
• Question 3

Which of the options below is the correct common ratio for this sequence?

1√5, 5, 5√5, ...

√5
EDDIE SAYS
We need to divide two adjacent terms to find the common ratio here: 5√5 ÷ 5 = √5 Don't forget to make your life easier, and avoid dividing by a surd if you can avoid it.
• Question 4

Match each sequence below to its correct common ratio.

## Column B

2, 2√2, 4, 4√2, ...
√2
9, 3√3, 3, √3, ...
√3
-10√2, -40, -80√2, .....
2√2
EDDIE SAYS
We can actually match these easily using a bit of logic. Which sequence involves √3? Out of the other two, which sequence is demonstrating the most significant changes between the terms? Match the highest ratio with the sequence showing the biggest changes. Did those tips help?
• Question 5

What is the common ratio for this sequence?

32√6, 16√6, 8√6, 4√6, ...

0.5
1/2
EDDIE SAYS
We don't even need to think about surds for this one, as the surd isn't changing in each term. Therefore, we can just ignore it when we're finding the common ratio. Each term is half of the previous one, so we are multiplying by 0.5 or 1/2. Did you type that ratio in accurately and without any spaces?
• Question 6

Consider the sequence below:

√2, 2, 2√2, ...

EDDIE SAYS
We just need to extend our sequence here until we reach an answer of 16. We can find the common ratio by dividing adjacent terms: 2√2 ÷ 2 = √2 Let's apply this ratio to the upcoming terms: 4th: 2√2 × √2 = 4 5th: 4 × √2 = 4√2 6th: 4√2 × √2 = 8 7th: 8 × √2 = 8√2 8th: 8√2 × √2 = 16 Do you agree?
• Question 7

What is the ninth term in this sequence?

12, 6√2, 6, 3√2, ...

0.75
3/4
EDDIE SAYS
Step 1: Find the common ratio by dividing two adjacent terms. 6√2 ÷ 12 = 1/√2 Step 2: Use this ratio to find the 9th term. 5th: 3√2 × 1/√2 = 3 6th: 3 × 1/√2 = 3/√2 7th: 3/√2 × 1/√2 = 3/2 8th: 3/2 × 1/√2 = 3/(2√2) 9th: 3/(2√2) × 1/√2 = 3/4 or 0.75 Phew, there were some tricky surds to work with there! Take your time and follow this method through carefully.
• Question 8

One of the terms in the sequence below is incorrect:

2√2, 4, 4√2, 8√2, ...

Which one is it?

8√2
EDDIE SAYS
Let's start with our two easiest terms to work with to find our ratio: 4√2 ÷ 4 = √2 If we assume that the common ratio here is √2, we need to find which two terms do not have this common ratio. 2√2 × √2 = 4 (matches) 4 × √2 = 4√2 (matches) 4√2 × √2 = 8 (does not match) So the final term (8√2) is incorrect, as this term should be 8.
• Question 9

Consider the sequence below:

-6, 12√3, -72, 144√3, ...

The common ratio for this sequence can be written in the form a√b.

Find the value of a and b.

EDDIE SAYS
We need to find our common ratio. Let's make our life easier and use the first two terms here, as the numbers are easiest to work with. 12√3 ÷ -6 = -2√3 So a = -2 and b = 3. Did you type those numbers in correctly, and did you remember the negative sign with a?
• Question 10

True or false?

The sequence 8, 4√2, 4, ... will never become negative.

True
EDDIE SAYS
Let's work out our common ratio here: 4√2 ÷ 8 = 1/√2 If we keep on multiplying a positive number by a positive number, will it ever become negative? Unless we introduce a negative sign through our ratio, then the sequence will never become negative. Congratulations! You have completed this very challenging activity.
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