Rdcarroll0412idn
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Shayna, Jamal, and Anjali are finding the 2nd differences for the sequence with the formula [tex]a_n = n^2 - 3[/tex].

Shayna says the 2nd differences are a constant value of 5.
Jamal says the 2nd differences are a constant value of 7.
Anjali says the 2nd differences are a constant value of 2.

Is Shayna, Jamal, or Anjali correct in finding the 2nd differences?

A. Jamal is correct. Shayna calculated based on the wrong terms, and Anjali subtracted too many times.
B. Anjali is correct. Jamal and Shayna both calculated 1st differences.
C. Anjali is correct because the polynomial is a degree of 2.
D. Shayna is correct. Jamal used the wrong terms, and Anjali subtracted too many times.

Sagot :

GinnyAnsweridn
To determine who is correct among Shayna, Jamal, and Anjali about the second differences of the sequence defined by [tex]\(a_n = n^2 - 3\)[/tex], we need to follow a step-by-step process to calculate the terms, first differences, and second differences.

Step 1: Calculate the first few terms of the sequence.

The sequence is given by [tex]\(a_n = n^2 - 3\)[/tex]. Let's calculate the first few terms:
- For [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 1^2 - 3 = 1 - 3 = -2 \][/tex]
- For [tex]\(n = 2\)[/tex]:
[tex]\[ a_2 = 2^2 - 3 = 4 - 3 = 1 \][/tex]
- For [tex]\(n = 3\)[/tex]:
[tex]\[ a_3 = 3^2 - 3 = 9 - 3 = 6 \][/tex]
- For [tex]\(n = 4\)[/tex]:
[tex]\[ a_4 = 4^2 - 3 = 16 - 3 = 13 \][/tex]

So, the terms are [tex]\([-2, 1, 6, 13]\)[/tex].

Step 2: Calculate the first differences between consecutive terms.

The first differences are calculated as follows:
- Between [tex]\(a_2\)[/tex] and [tex]\(a_1\)[/tex]:
[tex]\[ 1 - (-2) = 1 + 2 = 3 \][/tex]
- Between [tex]\(a_3\)[/tex] and [tex]\(a_2\)[/tex]:
[tex]\[ 6 - 1 = 5 \][/tex]
- Between [tex]\(a_4\)[/tex] and [tex]\(a_3\)[/tex]:
[tex]\[ 13 - 6 = 7 \][/tex]

So, the first differences are [tex]\([3, 5, 7]\)[/tex].

Step 3: Calculate the second differences between the first differences.

The second differences are calculated as follows:
- Between the second first difference and the first first difference:
[tex]\[ 5 - 3 = 2 \][/tex]
- Between the third first difference and the second first difference:
[tex]\[ 7 - 5 = 2 \][/tex]

So, the second differences are [tex]\([2, 2]\)[/tex].

Step 4: Determine the constant value of the second differences.

Since the second differences are all [tex]\(2\)[/tex], the constant value is [tex]\(2\)[/tex].

Based on our calculations, Anjali is correct because the second differences are indeed a constant value of 2, which aligns with the characteristics of a polynomial of degree 2. Thus, Anjali correctly identified the second differences, while Shayna and Jamal made incorrect calculations.

So, the correct statement is:
Anjali is correct because the polynomial is a degree of 2.
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