To determine who is correct in finding the second differences for the sequence defined by the formula [tex]\( a_n = n^2 - 3 \)[/tex], let's go through the process of calculating the first and second differences step by step.
1. Calculate the first few terms of the sequence:
[tex]\[
a_1 = 1^2 - 3 = -2
\][/tex]
[tex]\[
a_2 = 2^2 - 3 = 1
\][/tex]
[tex]\[
a_3 = 3^2 - 3 = 6
\][/tex]
[tex]\[
a_4 = 4^2 - 3 = 13
\][/tex]
2. Calculate the first differences (difference between consecutive terms):
[tex]\[
\Delta a_1 = a_2 - a_1 = 1 - (-2) = 3
\][/tex]
[tex]\[
\Delta a_2 = a_3 - a_2 = 6 - 1 = 5
\][/tex]
[tex]\[
\Delta a_3 = a_4 - a_3 = 13 - 6 = 7
\][/tex]
Thus, the first differences are [tex]\( \{3, 5, 7\} \)[/tex].
3. Calculate the second differences (difference between consecutive first differences):
[tex]\[
\Delta^2 a_1 = \Delta a_2 - \Delta a_1 = 5 - 3 = 2
\][/tex]
[tex]\[
\Delta^2 a_2 = \Delta a_3 - \Delta a_2 = 7 - 5 = 2
\][/tex]
Thus, the second differences are [tex]\( \{2, 2\} \)[/tex], which are indeed constant.
Given these calculations:
- Shayna says the second differences are 5.
- Jamal says the second differences are 7.
- Anjali says the second differences are 2.
Since the calculated second differences are [tex]\( \{2, 2\} \)[/tex]:
- Anjali is correct because the second differences are indeed a constant value of 2.
Therefore, the correct conclusion is:
- Anjali is correct. Jamal and Shayna both calculated 1st differences.
