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Explore Sequences Involving Surds

In this worksheet, students will explore sequences that involve surds.

'Explore Sequences Involving Surds' worksheet

Key stage:  KS 4

Curriculum topic:  

Curriculum subtopic:  

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

If you are aiming for the top grades, you need to be able to tie together sequences with 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.

For example, the sequence 2, 4, 8, 16 is generated by multiplying by 2.

 

How to find the common ratio.

This is surprisingly easy, all you have to do is to divide one term by the previous term, this will always give the common ratio.

 

Example: Continue the sequence √3, 3, 3√3, 9

Step 1: Find the common ratio.

I'm going to pick the third and second terms for this (just because the numbers are nice - you can pick whichever ones you want) to find the common ratio

3√3 ÷ 3 = √3

So my common ratio is √3

Step 2: Continue the sequence.

I now just need to multiply the fourth term by √3 to get the fifth term - 9 x √3 = 9√3

Then I can multiply this fifth term by √3 to get the sixth term - 9√3 x √3 = 9 x 3 = 27

Sequences involving surds will almost certainly be which type of sequence?

What is the next term in this sequence?

4, 4√3, 12, 12√3

Which of these options is the correct common ratio for the sequence 1√5,5,5√5?

5

√5

÷ √5

Match the sequence to the correct common ratio.

Column A

Column B

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

What is the common ratio for the sequence...

32√6, 16√6, 8√6, 4√6

In the sequence √2, 2, 2√2...

What is the 9th term in this sequence?

12, 6√2, 6, 3√2...

One of the terms in this sequence is incorrect, can you tell me which one it is?

 

2√2, 4, 4√2, 8√2

1st term

2nd term

3rd term

4th term

The common ratio for this sequence can be written as a√b.

 

-6, 12√3, -72, 144√3

 

Find the value of a and b

1st term

2nd term

3rd term

4th term

True or False.

8, 4√2, 4... will never be negative.

True

False

  • Question 1

Sequences involving surds will almost certainly be which type of sequence?

CORRECT ANSWER
geometric
EDDIE SAYS
While it is possible to have linear sequences involving surds, they are incredibly easy to do. Adding surds will therefore not be tested using sequences. The type of sequence that will challenge at this top-level will be geometric.
  • Question 2

What is the next term in this sequence?

4, 4√3, 12, 12√3

CORRECT ANSWER
36
EDDIE SAYS
Remember to find the common ratio. 4√3 ÷ 4 = √3 Then we can multiply 12√3 by √3 to find the next term
  • Question 3

Which of these options is the correct common ratio for the sequence 1√5,5,5√5?

CORRECT ANSWER
√5
EDDIE SAYS
As we mentioned, just divide any two adjacent terms to find the common ratio. Don't forget to make your life easier and don't do it so you have to divide by a surd if you can avoid it.
  • Question 4

Match the sequence to the correct common ratio.

CORRECT ANSWER

Column A

Column B

2, 2√2, 4, 4√2
√2
9, 3√3, 3, √3
÷ √3
-10√2, -40, -80√2
2√2
EDDIE SAYS
You can actually do this by using a bit of logic. Which one involves √3? Which one is changing the most out of the other two?
  • Question 5

What is the common ratio for the sequence...

32√6, 16√6, 8√6, 4√6

CORRECT ANSWER
0.5
1/2
EDDIE SAYS
You don't even need to think about surds for this one. As the surd isn't changing, 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 (1/2)
  • Question 6

In the sequence √2, 2, 2√2...

CORRECT ANSWER
EDDIE SAYS
We just need to extend our sequence here, the common ratio is √2 so we just keep multiplying by this until we get to 16.
  • Question 7

What is the 9th term in this sequence?

12, 6√2, 6, 3√2...

CORRECT ANSWER
0.75
3/4
EDDIE SAYS
Step 1: Find the common ratio (1/√2) Step 2: Use this to find the 9th term.
  • Question 8

One of the terms in this sequence is incorrect, can you tell me which one it is?

 

2√2, 4, 4√2, 8√2

CORRECT ANSWER
4th term
EDDIE SAYS
Did you spot it? If you found the common ratio (√2), you could just use this to generate the correct sequence, it's then easy to spot which one is wrong.
  • Question 9

The common ratio for this sequence can be written as a√b.

 

-6, 12√3, -72, 144√3

 

Find the value of a and b

CORRECT ANSWER
EDDIE SAYS
Remember what I said about making your life easy. I'd choose the first two terms so my numbers are smaller.
  • Question 10

True or False.

8, 4√2, 4... will never be negative.

CORRECT ANSWER
True
EDDIE SAYS
If you worked out our common ratio, you should have 1/√2. If you keep on multiplying a positive number by a positive number, will it ever become negative?
---- OR ----

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