When numbers are raised to power of 4

 Let's continue exploring the difference of consecutive numbers raised to the power of 4 for \( n = 1 \) through \( n = 10 \). As derived earlier, the difference between \( (n+1)^4 \) and \( n^4 \) is given by:

\[

(n+1)^4 - n^4 = 4n^3 + 6n^2 + 4n + 1

\]

We'll calculate these differences for \( n = 1 \) to \( n = 10 \):

- For \( n = 1 \):

  \[

  4(1)^3 + 6(1)^2 + 4(1) + 1 = 4 + 6 + 4 + 1 = 15

  \]

- For \( n = 2 \):

  \[

  4(2)^3 + 6(2)^2 + 4(2) + 1 = 32 + 24 + 8 + 1 = 65

  \]

- For \( n = 3 \):

  \[

  4(3)^3 + 6(3)^2 + 4(3) + 1 = 108 + 54 + 12 + 1 = 175

  \]

- For \( n = 4 \):

  \[

  4(4)^3 + 6(4)^2 + 4(4) + 1 = 256 + 96 + 16 + 1 = 369

  \]

- For \( n = 5 \):

  \[

  4(5)^3 + 6(5)^2 + 4(5) + 1 = 500 + 150 + 20 + 1 = 671

  \]

- For \( n = 6 \):

  \[

  4(6)^3 + 6(6)^2 + 4(6) + 1 = 864 + 216 + 24 + 1 = 1105

  \]

- For \( n = 7 \):

  \[

  4(7)^3 + 6(7)^2 + 4(7) + 1 = 1372 + 294 + 28 + 1 = 1695

  \]

- For \( n = 8 \):

  \[

  4(8)^3 + 6(8)^2 + 4(8) + 1 = 2048 + 384 + 32 + 1 = 2465

  \]

- For \( n = 9 \):

  \[

  4(9)^3 + 6(9)^2 + 4(9) + 1 = 2916 + 486 + 36 + 1 = 3439

  \]

- For \( n = 10 \):

  \[

  4(10)^3 + 6(10)^2 + 4(10) + 1 = 4000 + 600 + 40 + 1 = 4641

  \]

Thus, the sequence of differences between consecutive fourth powers from \( n = 1 \) to \( n = 10 \) is:

\[

15, 65, 175, 369, 671, 1105, 1695, 2465, 3439, 4641

\]

Now, let's calculate the differences between consecutive terms:

- \( 65 - 15 = 50 \)

- \( 175 - 65 = 110 \)

- \( 369 - 175 = 194 \)

- \( 671 - 369 = 302 \)

- \( 1105 - 671 = 434 \)

- \( 1695 - 1105 = 590 \)

- \( 2465 - 1695 = 770 \)

- \( 3439 - 2465 = 974 \)

- \( 4641 - 3439 = 1202 \)

The differences are: \( 50, 110, 194, 302, 434, 590, 770, 974, 1202 \).

Next, let's look at the second differences (the differences of the differences):

- \( 110 - 50 = 60 \)

- \( 194 - 110 = 84 \)

- \( 302 - 194 = 108 \)

- \( 434 - 302 = 132 \)

- \( 590 - 434 = 156 \)

- \( 770 - 590 = 180 \)

- \( 974 - 770 = 204 \)

- \( 1202 - 974 = 228 \)

The second differences are: \( 60, 84, 108, 132, 156, 180, 204, 228 \), which increase by 24 at each step.

### Observations:

- The first differences between consecutive terms (\( 15, 65, 175, \dots \)) grow by increasing amounts.

- The second differences (\( 60, 84, 108, 132, \dots \)) follow a linear pattern, increasing by 24 at each step.

  

This shows a clear structure in the differences for powers of 4, just like we saw with powers of 3, though it's a bit more complex, with the second differences being linear.