Let's determine the 2nd differences for the sequence given by the formula [tex]\( a_n = n^2 - 3 \)[/tex].
Step 1: Generate the first few terms of the sequence.
We calculate the values of [tex]\( a_n \)[/tex] for the first five terms, where [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
- [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]
- [tex]\( a_5 = 5^2 - 3 = 22 \)[/tex]
So, the five terms of the sequence are: [tex]\[ -2, 1, 6, 13, 22 \][/tex]
Step 2: Calculate the 1st differences (the differences between consecutive terms).
- The difference between [tex]\( a_2 \)[/tex] and [tex]\( a_1 \)[/tex]: [tex]\( 1 - (-2) = 3 \)[/tex]
- The difference between [tex]\( a_3 \)[/tex] and [tex]\( a_2 \)[/tex]: [tex]\( 6 - 1 = 5 \)[/tex]
- The difference between [tex]\( a_4 \)[/tex] and [tex]\( a_3 \)[/tex]: [tex]\( 13 - 6 = 7 \)[/tex]
- The difference between [tex]\( a_5 \)[/tex] and [tex]\( a_4 \)[/tex]: [tex]\( 22 - 13 = 9 \)[/tex]
So, the 1st differences are: [tex]\[ 3, 5, 7, 9 \][/tex]
Step 3: Calculate the 2nd differences (the differences between consecutive 1st differences).
- The difference between the second and first 1st differences: [tex]\( 5 - 3 = 2 \)[/tex]
- The difference between the third and second 1st differences: [tex]\( 7 - 5 = 2 \)[/tex]
- The difference between the fourth and third 1st differences: [tex]\( 9 - 7 = 2 \)[/tex]
So, the 2nd differences are: [tex]\[ 2, 2, 2 \][/tex]
Notice that the 2nd differences are constant.
Given these calculations, we see that Anjali is correct. She correctly identified that the 2nd differences for the sequence are a constant value of 2. Hence, Anjali is correct because the sequence [tex]\( a_n = n^2 - 3 \)[/tex] is a quadratic polynomial, which always results in constant 2nd differences.
