complete these series,1,8,27,64,___,____

a n = n ^ 3

1 ^ 3 = 1

2 ^ 3 = 8

3 ^ 3 = 27

4 ^ 3 = 64

5 ^ 3 = 125

6 ^ 3 = 216

To complete the series 1, 8, 27, 64, we need to identify the pattern or rule that governs the sequence. Once we have the pattern, we can apply it to find the missing numbers.

Let's take a look at the differences between consecutive terms:

8 - 1 = 7
27 - 8 = 19
64 - 27 = 37

The differences do not follow a linear pattern. Let's examine the differences between these differences:

19 - 7 = 12
37 - 19 = 18

It seems like the differences between the differences are decreasing by 6 each time. However, the differences between the differences are not consistent enough for us to determine the pattern.

Alternatively, let's try raising the numbers to different powers:

1^3 = 1
2^3 = 8
3^3 = 27
4^3 = 64

We can see that each term in the series is the cube of a natural number. Therefore, the missing numbers in the series are likely to be the cubes of the subsequent natural numbers.

Applying this pattern, we find that the missing numbers are:

5^3 = 125
6^3 = 216

So, the completed series is 1, 8, 27, 64, 125, 216.