Shayna, Jamal, and Anjali are finding the 2nd differences for the sequence with the formula an=n2−3 . 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? (1 point) Responses Jamal is correct. Shayna calculated based on the wrong terms, and Anjali subtracted too many times. Jamal is correct. Shayna calculated based on the wrong terms, and Anjali subtracted too many times. Shayna is correct. Jamal used the wrong terms, and Anjali subtracted too many times. Shayna is correct. Jamal used the wrong terms, and Anjali subtracted too many times. Anjali is correct because the polynomial is a degree of 2. Anjali is correct because the polynomial is a degree of 2. Anjali is correct. Jamal and Shayna both calculated 1st differences.

Question

GPT-4o mini · Answer

To determine who is correct about the second differences of the sequence defined by the formula \( a_n = n^2 - 3 \), we will first compute the terms of the sequence and then find the first and second differences.

Calculate the terms of the sequence:
- For \( n = 1 \): \( a_1 = 1^2 - 3 = 1 - 3 = -2 \)
- For \( n = 2 \): \( a_2 = 2^2 - 3 = 4 - 3 = 1 \)
- For \( n = 3 \): \( a_3 = 3^2 - 3 = 9 - 3 = 6 \)
- For \( n = 4 \): \( a_4 = 4^2 - 3 = 16 - 3 = 13 \)
- For \( n = 5 \): \( a_5 = 5^2 - 3 = 25 - 3 = 22 \)
The sequence is: \( -2, 1, 6, 13, 22 \).
Find the first differences:
- \( a_2 - a_1 = 1 - (-2) = 3 \)
- \( a_3 - a_2 = 6 - 1 = 5 \)
- \( a_4 - a_3 = 13 - 6 = 7 \)
- \( a_5 - a_4 = 22 - 13 = 9 \)
The first differences are: \( 3, 5, 7, 9 \).
Find the second differences:
- \( 5 - 3 = 2 \)
- \( 7 - 5 = 2 \)
- \( 9 - 7 = 2 \)
The second differences are constant: \( 2, 2, 2 \).

Since we found that the second differences are all 2, that means Anjali is correct. The correct response is:

Anjali is correct because the polynomial is a degree of 2.