Question

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

Answer 1

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

`1xx3=3`

Lets experiment with that for a moment!

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

`3xx3=9` which is correct.

Ok. lets apply the same rule yet again!

`9xx3=27`

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

`27xx3=81`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This works so the missing number is 27

Answer 2

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:

`color(blue)(1), 3, 9, x, 81`

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

`color(blue)(2), 6, x-9, 81-x`

Write out the sequence of differences of those differences:

`color(blue)(4), x-15, 90-2x`

Write out the sequence of differences of those differences:

`color(blue)(x-19), 105-3x`

If this is a constant sequence then we must have:

`x-19 = 105-3x`

Hence:

`4x = 124`

So:

`x = 31`

Note that `x-19 = 31-19=color(blue)(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)`

`color(white)(a_n) = 2n^3 - 10n^2 + 18n - 9`