Solved on Jan 18, 2024

Which of the following statements about quadratic number patterns is correct? $A. \text{The first difference is constant.}$ $B. \text{The third difference is greater than zero.}$

STEP 1

Assumptions
1. A quadratic number pattern is a sequence of numbers where the $n$ -th term can be represented as $an^2 + bn + c$ , where $a$ , $b$ , and $c$ are constants.
2. The first difference of a sequence is the set of differences between consecutive terms.
3. The second difference of a sequence is the set of differences between consecutive first differences.
4. The third difference of a sequence is the set of differences between consecutive second differences.

STEP 2

We need to understand what a constant difference implies for a sequence. A constant first difference indicates a linear pattern, while a constant second difference indicates a quadratic pattern.

STEP 3

To determine whether the first difference is constant, consider the general form of a quadratic sequence $an^2 + bn + c$ and calculate the first difference.

STEP 4

Calculate the $n$ -th term and the $(n+1)$ -th term of the sequence:
$T_n = an^2 + bn + c$ $T_{n+1} = a(n+1)^2 + b(n+1) + c$

STEP 5

Expand the $(n+1)$ -th term:
$T_{n+1} = a(n^2 + 2n + 1) + bn + b + c$ $T_{n+1} = an^2 + 2an + a + bn + b + c$

STEP 6

Calculate the first difference by subtracting the $n$ -th term from the $(n+1)$ -th term:
$D_1 = T_{n+1} - T_n$ $D_1 = (an^2 + 2an + a + bn + b + c) - (an^2 + bn + c)$

STEP 7

Simplify the first difference:
$D_1 = an^2 + 2an + a + bn + b + c - an^2 - bn - c$ $D_1 = 2an + a + b$

STEP 8

Notice that the first difference $D_1$ is not constant because it depends on $n$ . Therefore, statement A is incorrect.

STEP 9

To determine whether the second difference is constant, calculate the first difference for the $(n+2)$ -th term:
$T_{n+2} = a(n+2)^2 + b(n+2) + c$ $T_{n+2} = a(n^2 + 4n + 4) + bn + 2b + c$ $T_{n+2} = an^2 + 4an + 4a + bn + 2b + c$

STEP 10

Calculate the first difference between the $(n+1)$ -th and $(n+2)$ -th terms:
$D_1' = T_{n+2} - T_{n+1}$ $D_1' = (an^2 + 4an + 4a + bn + 2b + c) - (an^2 + 2an + a + bn + b + c)$

STEP 11

Simplify the first difference $D_1'$ :
$D_1' = an^2 + 4an + 4a + bn + 2b + c - an^2 - 2an - a - bn - b - c$ $D_1' = 2an + 3a + b$

STEP 12

Now calculate the second difference by subtracting the first difference $D_1$ from the first difference $D_1'$ :
$D_2 = D_1' - D_1$ $D_2 = (2an + 3a + b) - (2an + a + b)$

STEP 13

Simplify the second difference:
$D_2 = 2an + 3a + b - 2an - a - b$ $D_2 = 2a$

STEP 14

Notice that the second difference $D_2$ is constant because it does not depend on $n$ . Therefore, statement D is correct.

STEP 15

Since we have already found a correct statement, we do not need to check statements B and C. However, for completeness, let's briefly analyze them.

STEP 16

Statement B suggests that the third difference is greater than zero. Since the second difference is constant, the third difference, which would be the difference of constant second differences, would be zero. Therefore, statement B is incorrect.

STEP 17

Statement C suggests that the difference between terms is always positive. This is not necessarily true for a quadratic sequence, as the terms can increase and then decrease (or vice versa), depending on the sign and magnitude of $a$ . Therefore, statement C is also incorrect.
The correct statement about quadratic number patterns is: D. The second difference is constant.

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Which of the following statements about quadratic number patterns is correct? $A. \text{The first difference is constant.}$ $B. \text{The third difference is greater than zero.}$

STEP 1

STEP 2

We need to understand what a constant difference implies for a sequence. A constant first difference indicates a linear pattern, while a constant second difference indicates a quadratic pattern.

STEP 3

To determine whether the first difference is constant, consider the general form of a quadratic sequence $an^2 + bn + c$ and calculate the first difference.

STEP 4

Calculate the $n$ -th term and the $(n+1)$ -th term of the sequence:
$T_n = an^2 + bn + c$ $T_{n+1} = a(n+1)^2 + b(n+1) + c$

STEP 5

Expand the $(n+1)$ -th term:
$T_{n+1} = a(n^2 + 2n + 1) + bn + b + c$ $T_{n+1} = an^2 + 2an + a + bn + b + c$

STEP 6

Calculate the first difference by subtracting the $n$ -th term from the $(n+1)$ -th term:
$D_1 = T_{n+1} - T_n$ $D_1 = (an^2 + 2an + a + bn + b + c) - (an^2 + bn + c)$

STEP 7

Simplify the first difference:
$D_1 = an^2 + 2an + a + bn + b + c - an^2 - bn - c$ $D_1 = 2an + a + b$

STEP 8

Notice that the first difference $D_1$ is not constant because it depends on $n$ . Therefore, statement A is incorrect.

STEP 9

To determine whether the second difference is constant, calculate the first difference for the $(n+2)$ -th term:
$T_{n+2} = a(n+2)^2 + b(n+2) + c$ $T_{n+2} = a(n^2 + 4n + 4) + bn + 2b + c$ $T_{n+2} = an^2 + 4an + 4a + bn + 2b + c$

STEP 10

Calculate the first difference between the $(n+1)$ -th and $(n+2)$ -th terms:
$D_1' = T_{n+2} - T_{n+1}$ $D_1' = (an^2 + 4an + 4a + bn + 2b + c) - (an^2 + 2an + a + bn + b + c)$

STEP 11

Simplify the first difference $D_1'$ :
$D_1' = an^2 + 4an + 4a + bn + 2b + c - an^2 - 2an - a - bn - b - c$ $D_1' = 2an + 3a + b$

STEP 12

Now calculate the second difference by subtracting the first difference $D_1$ from the first difference $D_1'$ :
$D_2 = D_1' - D_1$ $D_2 = (2an + 3a + b) - (2an + a + b)$

STEP 13

Simplify the second difference:
$D_2 = 2an + 3a + b - 2an - a - b$ $D_2 = 2a$

STEP 14

Notice that the second difference $D_2$ is constant because it does not depend on $n$ . Therefore, statement D is correct.

STEP 15

Since we have already found a correct statement, we do not need to check statements B and C. However, for completeness, let's briefly analyze them.

STEP 16

Statement B suggests that the third difference is greater than zero. Since the second difference is constant, the third difference, which would be the difference of constant second differences, would be zero. Therefore, statement B is incorrect.

STEP 17

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STEP 1

STEP 2

We need to understand what a constant difference implies for a sequence. A constant first difference indicates a linear pattern, while a constant second difference indicates a quadratic pattern.

STEP 3

To determine whether the first difference is constant, consider the general form of a quadratic sequence an2+bn+can^2 + bn + can2+bn+c and calculate the first difference.

STEP 4

Calculate the nnn-th term and the (n+1)(n+1)(n+1)-th term of the sequence:Tn=an2+bn+cT_n = an^2 + bn + cTn​=an2+bn+c Tn+1=a(n+1)2+b(n+1)+cT_{n+1} = a(n+1)^2 + b(n+1) + cTn+1​=a(n+1)2+b(n+1)+c

STEP 5

Expand the (n+1)(n+1)(n+1)-th term:Tn+1=a(n2+2n+1)+bn+b+cT_{n+1} = a(n^2 + 2n + 1) + bn + b + cTn+1​=a(n2+2n+1)+bn+b+c Tn+1=an2+2an+a+bn+b+cT_{n+1} = an^2 + 2an + a + bn + b + cTn+1​=an2+2an+a+bn+b+c

STEP 6

Calculate the first difference by subtracting the nnn-th term from the (n+1)(n+1)(n+1)-th term:D1=Tn+1−TnD_1 = T_{n+1} - T_nD1​=Tn+1​−Tn​ D1=(an2+2an+a+bn+b+c)−(an2+bn+c)D_1 = (an^2 + 2an + a + bn + b + c) - (an^2 + bn + c)D1​=(an2+2an+a+bn+b+c)−(an2+bn+c)

STEP 7

Simplify the first difference:D1=an2+2an+a+bn+b+c−an2−bn−cD_1 = an^2 + 2an + a + bn + b + c - an^2 - bn - cD1​=an2+2an+a+bn+b+c−an2−bn−c D1=2an+a+bD_1 = 2an + a + bD1​=2an+a+b

STEP 8

Notice that the first difference D1D_1D1​ is not constant because it depends on nnn. Therefore, statement A is incorrect.

STEP 9

STEP 10

STEP 11

Simplify the first difference D1′D_1'D1′​:D1′=an2+4an+4a+bn+2b+c−an2−2an−a−bn−b−cD_1' = an^2 + 4an + 4a + bn + 2b + c - an^2 - 2an - a - bn - b - cD1′​=an2+4an+4a+bn+2b+c−an2−2an−a−bn−b−c D1′=2an+3a+bD_1' = 2an + 3a + bD1′​=2an+3a+b

STEP 12

Now calculate the second difference by subtracting the first difference D1D_1D1​ from the first difference D1′D_1'D1′​:D2=D1′−D1D_2 = D_1' - D_1D2​=D1′​−D1​ D2=(2an+3a+b)−(2an+a+b)D_2 = (2an + 3a + b) - (2an + a + b)D2​=(2an+3a+b)−(2an+a+b)

STEP 13

Simplify the second difference:D2=2an+3a+b−2an−a−bD_2 = 2an + 3a + b - 2an - a - bD2​=2an+3a+b−2an−a−b D2=2aD_2 = 2aD2​=2a

STEP 14

Notice that the second difference D2D_2D2​ is constant because it does not depend on nnn. Therefore, statement D is correct.

STEP 15

Since we have already found a correct statement, we do not need to check statements B and C. However, for completeness, let's briefly analyze them.

STEP 16

Statement B suggests that the third difference is greater than zero. Since the second difference is constant, the third difference, which would be the difference of constant second differences, would be zero. Therefore, statement B is incorrect.

STEP 17

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To determine whether the first difference is constant, consider the general form of a quadratic sequence $an^2 + bn + c$ and calculate the first difference.

Calculate the $n$ -th term and the $(n+1)$ -th term of the sequence:
$T_n = an^2 + bn + c$ $T_{n+1} = a(n+1)^2 + b(n+1) + c$

Expand the $(n+1)$ -th term:
$T_{n+1} = a(n^2 + 2n + 1) + bn + b + c$ $T_{n+1} = an^2 + 2an + a + bn + b + c$

Calculate the first difference by subtracting the $n$ -th term from the $(n+1)$ -th term:
$D_1 = T_{n+1} - T_n$ $D_1 = (an^2 + 2an + a + bn + b + c) - (an^2 + bn + c)$

Simplify the first difference:
$D_1 = an^2 + 2an + a + bn + b + c - an^2 - bn - c$ $D_1 = 2an + a + b$

Notice that the first difference $D_1$ is not constant because it depends on $n$ . Therefore, statement A is incorrect.

Simplify the first difference $D_1'$ :
$D_1' = an^2 + 4an + 4a + bn + 2b + c - an^2 - 2an - a - bn - b - c$ $D_1' = 2an + 3a + b$

Now calculate the second difference by subtracting the first difference $D_1$ from the first difference $D_1'$ :
$D_2 = D_1' - D_1$ $D_2 = (2an + 3a + b) - (2an + a + b)$

Simplify the second difference:
$D_2 = 2an + 3a + b - 2an - a - b$ $D_2 = 2a$

Notice that the second difference $D_2$ is constant because it does not depend on $n$ . Therefore, statement D is correct.