Generating Terms of a Sequence

Find the common difference for 3, 7, 11, 15, 19...

Solution:

To start with you can subtract the second term from the first term, then check that the difference is constant by checking the difference between each of the terms.

$7-3=4$

$11-7=4$

$15-11=4$

$19-15=4$

Here you can see that each of the terms has a difference of 4, therefore the common difference is 4.

Find the common ratio for 5, 10, 20, 40...

Solution:

To find the common ratio you can divide one term by the previous term, then you can check that the difference between each term is constant;

$10÷5=2$

$20÷10=2$

$40÷20=2$

Here you can see that the common ratio is 2.

Use the term-to-term rule to find the next 3 terms of the following sequence; 6, 11, 16, 21...

Solution:

Find the common difference by calculating the difference between each term to ensure it is constant.

$11-6=5$

$16-11=5$

$21-16=5$

You can see here that the common difference is 5, since you know this you can now use this information to find the next terms of the sequence. This is done by adding 5 to the last term to find the next term.

$21+5=26$

$26+5=31$

$31+5=36$

Therefore, for the sequence; 6, 11, 16, 21... the common difference is 5 and the next three terms are 26, 31 and 36.

Let's look at the sequence; 10, 11, 12, 13, 14...

Solution:

To visually show how to identify the position of each of these terms we will put them into a table.

In order to create the formula, you need to consider how the terms are getting from the position to the term, for example, to get from position 1 to the term 10 you add 9. Again, to get from position 2 to term 11, you add 9 and so on. This means that the formula will be written as

$n+9$

Find the first three terms of the sequence where $2n+6$

Solution:

Since you need to find the first three terms of the sequence, you simply substitute the positions, 1, 2 and 3 into the formula to find the value of the terms;

$2\left(1\right)+6=8$

$2\left(2\right)+6=10$

$2\left(3\right)+6=12$

Therefore the first three terms of the sequence $2n+6$ are 8, 10, 12.

Find the first three terms of the sequence where $4n-2$ using the position to term rule.

Solution:

Since you need to find the first three terms of the sequence, you simply substitute the positions into the formula.

$4\left(1\right)-2=2$

$4\left(2\right)-2=6$

$4\left(3\right)-2=10$

Therefore the first three terms of the sequence $4n-2$ are 2, 6, 10.

Use the term-to-term rule to find the next 3 terms of the following sequence; 4, 12, 36, 108...

Solution:

Start by finding the constant between each other the terms:

$12÷4=3$

$36÷12=3$

$108÷36=3$

You can see here that the common ratio is 3, now you know this you can use this information to find the next terms of the sequence. This is done by multiplying the last term by 3 to find the next term;

$108\times 3=324$

$324\times 3=972$

$972\times 3=2916$

Therefore the next 3 terms of the sequence are 324, 972 and 2916.