To find the second differences of the sequence generated by the polynomial [tex]\( n^2 + n + 1 \)[/tex], we'll follow these steps:
### Step 1: Calculate the sequence values
First, evaluate the polynomial [tex]\( n^2 + n + 1 \)[/tex] for the first few integer 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]
Thus, the sequence values are:
[tex]\[ 3, 7, 13, 21, 31 \][/tex]
### Step 2: Calculate the first differences
Next, find the first differences between consecutive terms in the sequence.
First difference between [tex]\( 7 \)[/tex] and [tex]\( 3 \)[/tex]:
[tex]\[ 7 - 3 = 4 \][/tex]
First difference between [tex]\( 13 \)[/tex] and [tex]\( 7 \)[/tex]:
[tex]\[ 13 - 7 = 6 \][/tex]
First difference between [tex]\( 21 \)[/tex] and [tex]\( 13 \)[/tex]:
[tex]\[ 21 - 13 = 8 \][/tex]
First difference between [tex]\( 31 \)[/tex] and [tex]\( 21 \)[/tex]:
[tex]\[ 31 - 21 = 10 \][/tex]
Thus, the first differences are:
[tex]\[ 4, 6, 8, 10 \][/tex]
### Step 3: Calculate the second differences
Finally, find the second differences between consecutive terms in the first differences.
Second difference between [tex]\( 6 \)[/tex] and [tex]\( 4 \)[/tex]:
[tex]\[ 6 - 4 = 2 \][/tex]
Second difference between [tex]\( 8 \)[/tex] and [tex]\( 6 \)[/tex]:
[tex]\[ 8 - 6 = 2 \][/tex]
Second difference between [tex]\( 10 \)[/tex] and [tex]\( 8 \)[/tex]:
[tex]\[ 10 - 8 = 2 \][/tex]
Thus, the second differences are:
[tex]\[ 2, 2, 2 \][/tex]
### Conclusion
The second differences of the sequence generated by the polynomial [tex]\( n^2 + n + 1 \)[/tex] are:
[tex]\[ 2, 2, 2 \][/tex]
