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

Mar 8, 2017

$33 , 39$

#### 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: ${a}_{\text{n+1}} = {a}_{n} + 6$

We can test this on the ${4}^{t h}$ 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.

$\therefore {a}_{5} = 27 + 6 = 33$

and

${a}_{6} = 33 + 6 = 39$

Mar 10, 2017

$33 \text{ , } 39$

#### 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?

$1 , 3 , 5 , 7 , \ldots$ would be odd numbers.
$1 , 4 , 9 , 16 \ldots .$ would be square numbers.
$2 , 3 , 5 , 7 , 11 \ldots .$ would be prime numbers
$7 , 14 , 21 , 28 , \ldots .$ 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 ${n}^{\text{th}}$ term? Like ${T}_{n} = {n}^{2} / \left(n + 1\right)$?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In this case we have $9 , 15 , 21 , 27 \ldots . .$

You should see that the term-to-term rule is "add 6"
$\textcolor{w h i t e}{\ldots} \textcolor{red}{+ 6 \rightarrow} \text{ "color(red)(+6rarr)" "color(red)(+6rarr)" } \textcolor{red}{+ 6 \rightarrow}$
$9 , \text{ " 15," " 21," "27" }$

So, following this pattern gives the next 2 terms as $33 \mathmr{and} 39$