Sure, let's solve this step by step.
1. Define the polynomial function:
The given polynomial is [tex]\( n^2 + n + 1 \)[/tex].
2. Generate the first few terms of the sequence:
We will calculate this polynomial for the first five values of [tex]\( n \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[
1^2 + 1 + 1 = 3
\][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[
2^2 + 2 + 1 = 7
\][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[
3^2 + 3 + 1 = 13
\][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[
4^2 + 4 + 1 = 21
\][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[
5^2 + 5 + 1 = 31
\][/tex]
So, the terms of the sequence are: [tex]\([3, 7, 13, 21, 31]\)[/tex].
3. Calculate the first differences:
The first differences are obtained by subtracting each term from the next term:
- [tex]\(7 - 3 = 4\)[/tex]
- [tex]\(13 - 7 = 6\)[/tex]
- [tex]\(21 - 13 = 8\)[/tex]
- [tex]\(31 - 21 = 10\)[/tex]
Therefore, the first differences are: [tex]\([4, 6, 8, 10]\)[/tex].
4. Calculate the second differences:
The second differences are obtained by calculating the differences between consecutive first differences:
- [tex]\(6 - 4 = 2\)[/tex]
- [tex]\(8 - 6 = 2\)[/tex]
- [tex]\(10 - 8 = 2\)[/tex]
Therefore, the second differences are: [tex]\([2, 2, 2]\)[/tex].
Thus, the second differences of the sequence generated by the polynomial [tex]\( n^2 + n + 1 \)[/tex] are [tex]\([2, 2, 2]\)[/tex].
