Comparable to the easier-to-recognize series of squares: 1 4 9 16 25 36 etc.

Comparable to the easier-to-recognize series of squares: 1 4 9 16 25 36 etc.

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.

*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.

2 cubed = 8

3 cubed = 27

4 X4X4 = 64

5 cubed = 125

5X5X5