# How Do I Solve This Maths Pattern - 1 - 8 - 27 - ? - 125 - 216

## 5 Answers

Anonymous answered
1 - 8 - 27 - 64 - 125 - 216 Series of cubes. 1^3 = 1,  2^3 = 8,  3^3 = 27,  etc.
Comparable to the easier-to-recognize series of squares: 1 4 9 16 25 36 etc.
thanked the writer.
Joe Blow answered

Here is an odd thing about this question it is intended to show cubes, but mistakenly adds a second answer in 46.

1 - 8 - 27 - X - 125 - 216.  The best answer is 4^3.  (64) It is the simplest solution.

Here is why 46 also works.

The difference between 1 and 8 is 7.  The difference between 8 and 27 is 19.  Add the two differences to get 26.  Add the difference to the first number in the set.  1 + 26 = 27.  Move down one number and you get the same  pattern.  27-8=19.  46-27=19.  19+19+8=46.

The next step in formula is 19 + 79 + 27 = 125.

then 79 + 91 + 46 = 216.

(Difference + Difference + First number in set) = new number in the set.

The second answer could be eliminated by claiming that 1 and 8 have no way to be produced in the beginning, this however is does not matter because the instructions merely ask you to continue a pattern.  Both 64 and 46 continue patterns.

thanked the writer.
Oddman answered
For many series, if you have no clue as to how the series is developed, it is convenient to look at the way the differences change. We want to find differences at some level that are the same. If we call the unknown number "x", for "first differences," we have   8 - 1 = 7   27 - 8 = 19   x - 27   125 - x   216 - 125 = 91    (91 ≠ 19 ≠ 7, so the series is not linear) For second differences (differences between the numbers in the first-differences series), we have   19 - 7 = 12   (x - 27) - 19 = x - 46   (125 - x) - (x - 27) = 152 - 2x   91 - (125 - x) = x - 34    ((x-34) ≠ (x-46), so the series is not quadratic) For third differences (differences between the numbers in the second differences series), we have   (x - 46) - 12 = x - 58   (152 - 2x) - (x - 46) = 198 - 3x   (x - 34) - (152 - 2x) = 3x - 186    (The relations here are not obvious. We can see if one value of x will make these the same. Finding fourth differences will help with that.) Fourth differences are   (198 - 3x) - (x - 58) = 256 - 4x   (3x - 186) - (198 - 3x) = 6x - 384    (if both of these are zero (0), then third differences are the same. For 256-4x = 0, x = 256/4 = 64. For 6x-384 = 0, x = 384/6 = 64. It looks like we have an answer.) Third differences among the series elements can be made to be the same by the choice of 64 as the missing number. This makes the series a cubic series.
Actually,   1^3 = 1   2^3 = 8   3^3 = 27   4^3 = 64    (the missing number)   5^3 = 125   6^3 = 216 You will find that variations on this cubic series show up in a number of algebra problems. It will be convenient for you to learn these first few cubes, at least through 5^3 = 125.
##### You can also factor the numbers to gain a clue as to how the series evolves. 1=1, 8=2*2*2, 27=3*3*3, 125=5*5*5.
thanked the writer.
Anonymous answered
Its anynumber (cubed) man, 1 cubed = 1

2 cubed = 8

3 cubed = 27
4 X4X4 = 64
5 cubed = 125

5X5X5
thanked the writer.