Question

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?

Answer 1

To determine who is correct regarding the 2nd differences for the sequence defined by the formula \( a_n = n^2 - 3 \), we need to compute the first differences and then the second differences.

1. **Compute the values of \( a_n \)** for the first few values of \( n \):

- 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 we have the following sequence:
\[
a_1 = -2, \quad a_2 = 1, \quad a_3 = 6, \quad a_4 = 13
\]

2. **Compute the first differences** (\( \Delta a_n = a_{n+1} - a_n \)):
- \( \Delta a_1 = a_2 - a_1 = 1 - (-2) = 3 \)
- \( \Delta a_2 = a_3 - a_2 = 6 - 1 = 5 \)
- \( \Delta a_3 = a_4 - a_3 = 13 - 6 = 7 \)

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

3. **Compute the second differences** (\( \Delta^2 a_n = \Delta a_{n+1} - \Delta a_n \)):
- \( \Delta^2 a_1 = \Delta a_2 - \Delta a_1 = 5 - 3 = 2 \)
- \( \Delta^2 a_2 = \Delta a_3 - \Delta a_2 = 7 - 5 = 2 \)

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

Since the second differences are constant and equal to \( 2 \), **Anjali is correct** in stating that the second differences are a constant value of 2.