# What is the missing number in the pattern 1, 3, 9, __ , 81?

Jan 21, 2016

Same thing as others wrote but written slightly differently.

The missing number is 27

#### Explanation:

It is a matter of spotting the relationship if you can. If not obvious you will need to experiment to find what type of sequence it is; geometric or arithmetic.

Observe that there is a connection of 3 in the numbers 1,3,9 in that

$1 \times 3 = 3$

Lets experiment with that for a moment!

If we apply the same rule to 3 what will we get?

$3 \times 3 = 9$ which is correct.

Ok. lets apply the same rule yet again!

$9 \times 3 = 27$

Using 27, if we apply the rule once more and get 81 we have found the correct 'rule' for this sequence.

$27 \times 3 = 81$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This works so the missing number is 27

Apr 30, 2017

It would be $31$ if the sequence is defined by a cubic formula.

#### Explanation:

The normal assumption with this sequence is that it is arithmetic or geometric. The given numbers are consistent with a geometric sequence, resulting in the answer $27$.

However, note that we are not told that it is a geometric sequence, so there could be some other pattern.

For example, suppose it is a cubic sequence. Then taking the differences $3$ times will result in a constant sequence.

Let us write out the given sequence, with $x$ for the missing number:

$\textcolor{b l u e}{1} , 3 , 9 , x , 81$

Write out the sequence of differences between pairs of consecutive terms:

$\textcolor{b l u e}{2} , 6 , x - 9 , 81 - x$

Write out the sequence of differences of those differences:

$\textcolor{b l u e}{4} , x - 15 , 90 - 2 x$

Write out the sequence of differences of those differences:

$\textcolor{b l u e}{x - 19} , 105 - 3 x$

If this is a constant sequence then we must have:

$x - 19 = 105 - 3 x$

Hence:

$4 x = 124$

So:

$x = 31$

Note that $x - 19 = 31 - 19 = \textcolor{b l u e}{12}$

Then the cubic formula for a term of the sequence can be written:

a_n = color(blue)(1)/(0!)+color(blue)(2)/(1!)(n-1)+color(blue)(4)/(2!)(n-1)(n-2)+color(blue)(12)/(3!)(n-1)(n-2)(n-3)

$\textcolor{w h i t e}{{a}_{n}} = 2 {n}^{3} - 10 {n}^{2} + 18 n - 9$