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

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.
Anjali is correct. Jamal and Shayna both calculated 1st differences.

Jamal is correct. Shayna calculated based on the wrong terms, and Anjali subtracted too many times.

Answer 1

To determine who is correct among Shayna, Jamal, and Anjali, let's find the second differences of the sequence defined by the formula \( a_n = n^2 - 3 \).

1. Calculate the first few 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 \( n = 4: a_4 = 4^2 - 3 = 16 - 3 = 13 \)

   So the first few terms are: \(-2, 1, 6, 13\).

2. Calculate the first differences:

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

   The first differences are: \(3, 5, 7\).

3. Calculate the second differences:

   • First difference from \(3\) to \(5\): \(5 - 3 = 2\)
   • First difference from \(5\) to \(7\): \(7 - 5 = 2\)

   The second differences are constant and equal to: \(2, 2\), which means the constant second difference is \(2\).

From the calculations, we see that Anjali is correct with the constant second differences being \(2\).

Therefore, the correct response is: Anjali is correct because the polynomial is a degree of 2.