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.
