Shayna, Jamal, and Anjali are finding the 2nd differences for the sequence with the formula an=n2−3 .

Question

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.

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

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. Jamal and Shayna both calculated 1st differences.

Answer 1

To find the second differences of the sequence given by the formula \( a_n = n^2 - 3 \), we first need to calculate the first differences and then the second differences.

1. Calculate the first few terms of the sequence:

   For \( n = 1, 2, 3, 4 \):

   • \( a_1 = 1^2 - 3 = 1 - 3 = -2 \)
   • \( a_2 = 2^2 - 3 = 4 - 3 = 1 \)
   • \( a_3 = 3^2 - 3 = 9 - 3 = 6 \)
   • \( a_4 = 4^2 - 3 = 16 - 3 = 13 \)

   So, the first four terms are:

   • \( a_1 = -2 \)
   • \( a_2 = 1 \)
   • \( a_3 = 6 \)
   • \( a_4 = 13 \)

2. Calculate the first differences:

   The first differences are:

   • \( a_2 - a_1 = 1 - (-2) = 3 \)
   • \( a_3 - a_2 = 6 - 1 = 5 \)
   • \( a_4 - a_3 = 13 - 6 = 7 \)

   So, the first differences are:

   • \( 3, 5, 7 \).

3. Calculate the second differences:

   The second differences are:

   • \( 5 - 3 = 2 \)
   • \( 7 - 5 = 2 \)

   So, the second differences are constant and equal to \( 2 \).

Based on this calculation, Anjali is correct.

The response that accurately reflects this is: Anjali is correct because the polynomial is a degree of 2.