To find the second differences of the sequence defined by the polynomial [tex]\( n^2 + n + 1 \)[/tex], we will follow these steps:
1. Calculate the first few terms of the sequence:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[
1^2 + 1 + 1 = 1 + 1 + 1 = 3
\][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[
2^2 + 2 + 1 = 4 + 2 + 1 = 7
\][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[
3^2 + 3 + 1 = 9 + 3 + 1 = 13
\][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[
4^2 + 4 + 1 = 16 + 4 + 1 = 21
\][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[
5^2 + 5 + 1 = 25 + 5 + 1 = 31
\][/tex]
So, the first five terms of the sequence are: [tex]\( 3, 7, 13, 21, 31 \)[/tex].
2. Calculate the first differences of the sequence:
- Between the 1st and 2nd terms:
[tex]\[
7 - 3 = 4
\][/tex]
- Between the 2nd and 3rd terms:
[tex]\[
13 - 7 = 6
\][/tex]
- Between the 3rd and 4th terms:
[tex]\[
21 - 13 = 8
\][/tex]
- Between the 4th and 5th terms:
[tex]\[
31 - 21 = 10
\][/tex]
So, the first differences are: [tex]\( 4, 6, 8, 10 \)[/tex].
3. Calculate the second differences of the sequence:
- Between the 1st and 2nd first differences:
[tex]\[
6 - 4 = 2
\][/tex]
- Between the 2nd and 3rd first differences:
[tex]\[
8 - 6 = 2
\][/tex]
- Between the 3rd and 4th first differences:
[tex]\[
10 - 8 = 2
\][/tex]
So, the second differences are: [tex]\( 2, 2, 2 \)[/tex].
Therefore, the second differences of the sequence defined by the polynomial [tex]\( n^2 + n + 1 \)[/tex] are [tex]\( 2, 2, 2 \)[/tex].
