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

Question

𝑎
𝑛
=
𝑛
2
−
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

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.

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. Jamal and Shayna both calculated 1st differences.

Answer 1

To find the correct answer, we first need to calculate the sequence based on the formula \( a_n = n^2 - 3 \).

Let's compute 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 \( n = 4 \): \( a_4 = 4^2 - 3 = 16 - 3 = 13 \)
• For \( n = 5 \): \( a_5 = 5^2 - 3 = 25 - 3 = 22 \)

Now we have the sequence: \[ -2, 1, 6, 13, 22 \]

Next, we will calculate the first differences:

• \( 1 - (-2) = 3 \)
• \( 6 - 1 = 5 \)
• \( 13 - 6 = 7 \)
• \( 22 - 13 = 9 \)

The first differences are: \[ 3, 5, 7, 9 \]

Now we compute the second differences:

• \( 5 - 3 = 2 \)
• \( 7 - 5 = 2 \)
• \( 9 - 7 = 2 \)

The second differences are: \[ 2, 2, 2 \]

Since the second differences are constant at 2, Anjali is correct in finding the 2nd differences.

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