Question

Based on the pattern, what are the next two terms of the sequence 9, 15, 21, 27, ...?

Answer 1

`33, 39`

Explanation:

Let's look at the sequence term by term:

`a_1 = 9`
`a_2 = 15`
`a_3 = 21`

Notice that: `a_2 -a_1 =6` and `a_3 - a_2 =6`

We can deduce that: `a_"n+1" = a_n + 6`

We can test this on the `4^(th)` term: `a_4` should equal `21+6 = 27`

Since this checks out we can say that the sequence is an arithmetic progression with a common diference of 6.

`:. a_5 = 27+ 6 = 33`

and

`a_6 =33+6 = 39`

Answer 2

`33" , "39`

Explanation:

When you are presented with a sequence of numbers which you have to continue, there are different possibilities to consider.....

• Ask yourself whether the numbers are a specific type of number?

`1, 3, 5, 7, ...` would be odd numbers.
`1, 4, 9, 16 ....` would be square numbers.
`2,3, 5, 7, 11 ....` would be prime numbers
`7, 14, 21, 28, ....` would be multiples of 7.

If you recognise a that a certain type of number has been given you can easily fill in the next terms.

• Ask whether there is a term-to-term rule which you can use to get from one term to the next.
• This is often by adding or subtracting the same number each time, this gives an Arithmetic Sequence. (A.P.)
• It can be by multiplying or dividing by the same number each time, this gives a Geometric Sequence. (G.P.)

• Maybe adding on the previous term gives the next term. This is called a Fibonacci sequence.

• Has a rule been given for the `n^("th")` term? Like `T_n = n^2/(n+1)`?
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In this case we have `9, 15, 21,27 .....`

You should see that the term-to-term rule is "add 6"
`color(white)(...)color(red)(+6rarr)" "color(red)(+6rarr)" "color(red)(+6rarr)" "color(red)(+6rarr)`
`9," " 15," " 21," "27" "`

So, following this pattern gives the next 2 terms as `33 and 39`