Find the first five terms of the quadratic polynomial sequence with the formula a^n=3n^2-2. Calculate the 1st differences. Then, calculate the 2nd differences and show that they are constant. Include the constant value for the 2nd differences in your response.
To find the first five terms of the quadratic polynomial sequence, we substitute n = 1, 2, 3, 4, 5 into the formula a^n = 3n^2 - 2:
a^1 = 3(1)^2 - 2 = 3 - 2 = 1
a^2 = 3(2)^2 - 2 = 3(4) - 2 = 12 - 2 = 10
a^3 = 3(3)^2 - 2 = 3(9) - 2 = 27 - 2 = 25
a^4 = 3(4)^2 - 2 = 3(16) - 2 = 48 - 2 = 46
a^5 = 3(5)^2 - 2 = 3(25) - 2 = 75 - 2 = 73
The first five terms of the sequence are 1, 10, 25, 46, 73.
To calculate the 1st differences, we find the difference between consecutive terms:
10 - 1 = 9
25 - 10 = 15
46 - 25 = 21
73 - 46 = 27
The 1st differences are 9, 15, 21, 27.
To calculate the 2nd differences, we find the difference between consecutive 1st differences:
15 - 9 = 6
21 - 15 = 6
27 - 21 = 6
The 2nd differences are all 6.
Since the 2nd differences are constant, the sequence is a quadratic polynomial sequence.
To find the first five terms of the quadratic polynomial sequence with the formula a^n = 3n^2 - 2, we can substitute values for n and evaluate the expression.
When n = 1:
a^1 = 3(1^2) - 2
a^1 = 3 - 2
a^1 = 1
When n = 2:
a^2 = 3(2^2) - 2
a^2 = 3(4) - 2
a^2 = 12 - 2
a^2 = 10
When n = 3:
a^3 = 3(3^2) - 2
a^3 = 3(9) - 2
a^3 = 27 - 2
a^3 = 25
When n = 4:
a^4 = 3(4^2) - 2
a^4 = 3(16) - 2
a^4 = 48 - 2
a^4 = 46
When n = 5:
a^5 = 3(5^2) - 2
a^5 = 3(25) - 2
a^5 = 75 - 2
a^5 = 73
The first five terms of the quadratic polynomial sequence are: 1, 10, 25, 46, 73.
To calculate the 1st differences, we subtract each term from the next term:
10 - 1 = 9
25 - 10 = 15
46 - 25 = 21
73 - 46 = 27
The 1st differences are: 9, 15, 21, 27.
Now, let's calculate the 2nd differences by subtracting each 1st difference from the next 1st difference:
15 - 9 = 6
21 - 15 = 6
27 - 21 = 6
The 2nd differences are: 6, 6, 6.
As we can see, the 2nd differences are constant, with a value of 6.
To find the first five terms of the quadratic polynomial sequence with the formula a^n = 3n^2 - 2, we can substitute values for n and calculate the corresponding values of a.
Let's calculate the values of a for the first five terms:
- For n = 1: a^1 = 3(1)^2 - 2 = 3 - 2 = 1
- For n = 2: a^2 = 3(2)^2 - 2 = 3(4) - 2 = 12 - 2 = 10
- For n = 3: a^3 = 3(3)^2 - 2 = 3(9) - 2 = 27 - 2 = 25
- For n = 4: a^4 = 3(4)^2 - 2 = 3(16) - 2 = 48 - 2 = 46
- For n = 5: a^5 = 3(5)^2 - 2 = 3(25) - 2 = 75 - 2 = 73
So, the first five terms of the quadratic polynomial sequence are: 1, 10, 25, 46, 73.
Now let's calculate the 1st differences. The 1st difference between terms is the difference between consecutive terms in the sequence.
- 1st difference between 10 and 1: 10 - 1 = 9
- 1st difference between 25 and 10: 25 - 10 = 15
- 1st difference between 46 and 25: 46 - 25 = 21
- 1st difference between 73 and 46: 73 - 46 = 27
The 1st differences are: 9, 15, 21, 27.
Now let's calculate the 2nd differences. The 2nd difference between differences is the difference between consecutive 1st differences.
- 2nd difference between 15 and 9: 15 - 9 = 6
- 2nd difference between 21 and 15: 21 - 15 = 6
- 2nd difference between 27 and 21: 27 - 21 = 6
The 2nd differences are constant with a value of 6.
Therefore, the constant value for the 2nd differences is 6.