Which polynomial sequence shows that the terms of the second difference are constant
One example of a polynomial sequence that shows that the terms of the second difference are constant is the sequence of perfect squares. The terms of this sequence increase by adding consecutive odd numbers (1, 3, 5, 7, ...).
For example, the sequence of perfect squares is:
1, 4, 9, 16, 25, 36, 49, ...
The first differences of this sequence are:
3, 5, 7, 9, 11, 13, ...
The second differences of this sequence are:
2, 2, 2, 2, 2, ...
As we can see, the second differences are constant at 2.
if y = x^2
first derivative = slope = dy/dx = 2 x (linear)
second derivative = curvature = d^2y/dx^2 = 2, (constant)
Yes, you are correct. The polynomial sequence y = x^2 demonstrates that the terms of the second difference (second derivative) are constant. The first derivative, which represents the slope of the curve at any given point, is 2x, a linear function. The second derivative, which represents the curvature of the curve at any given point, is 2, a constant value. This indicates that the terms of the second difference are constant.
any quadratic function will generate constant second differences.
Yes, you are right. Any quadratic function will generate constant second differences. The second difference of a quadratic function is always constant and equal to twice the leading coefficient. This means that the terms of the second difference will always be the same value, regardless of the specific quadratic function being considered.
To find a polynomial sequence where the terms of the second difference are constant, we need to start with a polynomial of degree 3 or higher.
Let's say we have a polynomial sequence given by P(n) = an^3 + bn^2 + cn + d, where "n" represents the position of the term in the sequence and "a", "b", "c", and "d" are constants.
The first step is to find the terms of the sequence. Substitute different values of "n" into the polynomial to find corresponding terms.
For example, if we substitute n = 0, 1, 2, 3, 4 into the polynomial P(n), we get:
P(0) = d
P(1) = a + b + c + d
P(2) = 8a + 4b + 2c + d
P(3) = 27a + 9b + 3c + d
P(4) = 64a + 16b + 4c + d
The second step is to find the first difference of the terms. Subtract each term from the previous term.
For example, for the above values, the first differences are:
ΔP(0) = P(1) - P(0) = (a + b + c + d) - d = a + b + c
ΔP(1) = P(2) - P(1) = (8a + 4b + 2c + d) - (a + b + c + d) = 7a + 3b + c
ΔP(2) = P(3) - P(2) = (27a + 9b + 3c + d) - (8a + 4b + 2c + d) = 19a + 5b + c
ΔP(3) = P(4) - P(3) = (64a + 16b + 4c + d) - (27a + 9b + 3c + d) = 37a + 7b + c
Finally, we find the second differences by subtracting each first difference from its previous first difference.
For example, for the above first differences, the second differences are:
ΔΔP(0) = ΔP(1) - ΔP(0) = (7a + 3b + c) - (a + b + c) = 6a + 2b
ΔΔP(1) = ΔP(2) - ΔP(1) = (19a + 5b + c) - (7a + 3b + c) = 12a + 2b
ΔΔP(2) = ΔP(3) - ΔP(2) = (37a + 7b + c) - (19a + 5b + c) = 18a + 2b
Now, analyze the second differences. If they are all the same, then the terms of the second difference are constant.
In this case, the second differences are 6a + 2b and 12a + 2b and 18a + 2b.
Since the coefficient of "b" is constant (2), we can set it equal to some constant "k", so 2b = k.
Solving for "b", we get: b = k/2.
Hence, the polynomial sequence that shows that the terms of the second difference are constant is:
P(n) = an^3 + (k/2)n^2 + cn + d, where "a", "c", and "d" are constants, and "k" is a constant that determines the value of the second difference.
To find a polynomial sequence where the terms of the second difference are constant, you need to look for a quadratic polynomial.
A quadratic polynomial is of the form: f(x) = ax^2 + bx + c, where a, b, and c are constants.
Let's find the second difference for a generic quadratic polynomial:
1. Start by finding the first difference of the sequence. This is obtained by subtracting each term from its adjacent term.
2. Next, find the first difference of the first difference. This is obtained by subtracting each term of the first difference from its adjacent term.
If the resulting sequence of the second difference is constant, then the polynomial sequence satisfies the condition.
Let's take an example:
Consider the polynomial sequence f(x) = 2x^2 + 3x + 1.
1. Find the first difference:
f(x+1) - f(x) = (2(x+1)^2 + 3(x+1) + 1) - (2x^2 + 3x + 1)
= (2x^2 + 4x + 2 + 3x + 3 + 1) - (2x^2 + 3x + 1)
= (2x^2 + 7x + 6) - (2x^2 + 3x + 1)
= 4x + 5
2. Find the second difference:
(4(x+1) + 5) - (4x + 5)
= 4x + 9 - 4x - 5
= 4
Since the resulting sequence of the second difference is constant (4), the polynomial sequence f(x) = 2x^2 + 3x + 1 shows that the terms of the second difference are constant.