Question

Sheena Jamal and Angela Lee are finding the second difference of the sequence with the formula a n equals N2 - 3 Sheena says that the second difference are the constant value of five Jamal says the second difference are the constant value of seven generally says the second difference are a consonant value constant value of 2 is Shana Jamal or Angelie correct in finding the second difference

Answer 1

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

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 \( n = 4 \):
\[
a_4 = 4^2 - 3 = 16 - 3 = 13
\]
- For \( n = 5 \):
\[
a_5 = 5^2 - 3 = 25 - 3 = 22
\]

So, the sequence is: \(-2, 1, 6, 13, 22\).

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 \( a_4 - a_3 \):
\[
13 - 6 = 7
\]
- First difference \( a_5 - a_4 \):
\[
22 - 13 = 9
\]

The first differences are: \(3, 5, 7, 9\).

3. **Calculate the second differences:**
- Second difference \( (5 - 3) \):
\[
2
\]
- Second difference \( (7 - 5) \):
\[
2
\]
- Second difference \( (9 - 7) \):
\[
2
\]

The second differences are: \(2, 2, 2\).

Since the second difference is constant and equal to \(2\), we find that Angela Lee is correct in her assertion.

**Conclusion:** Angela Lee is correct; the second difference is a constant value of \(2\).