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 find the 2nd differences of the sequence given by the formula \( a_n = n^2 - 3 \), we can follow these steps:

1. **Calculate the first few terms of the sequence** using the formula \( a_n = n^2 - 3 \):

- 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 \( n = 4 \):
\[
a_4 = 4^2 - 3 = 16 - 3 = 13
\]

Thus, the first terms of the sequence are:
\[
a_1 = -2, \quad a_2 = 1, \quad a_3 = 6, \quad a_4 = 13
\]

2. **Calculate the first differences**:
- First difference \( a_2 - a_1 = 1 - (-2) = 3 \)
- First difference \( a_3 - a_2 = 6 - 1 = 5 \)
- First difference \( a_4 - a_3 = 13 - 6 = 7 \)

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

3. **Calculate the second differences**:
- Second difference \( 5 - 3 = 2 \)
- Second difference \( 7 - 5 = 2 \)

The second differences are constant and equal to:
\[
2, \quad 2
\]

From these calculations, we find that the second differences are indeed a constant value of **2**.

Therefore, **Anjali is correct** in finding the 2nd differences.