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

2 Answers
Jan 21, 2016

Answer:

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

Apr 30, 2017

Answer:

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#