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Shayna, Jamal, and Anjali are finding the second differences for the sequence with the formula [tex]a_n = n^2 - 3[/tex].

- Shayna says the second differences are a constant value of 5.
- Jamal says the second differences are a constant value of 7.
- Anjali says the second differences are a constant value of 2.

Is Shayna, Jamal, or Anjali correct in finding the second differences?

A. Jamal is correct. Shayna calculated based on the wrong terms, and Anjali subtracted too many times.
B. Anjali is correct. Jamal and Shayna both calculated first differences.
C. Shayna is correct. Jamal used the wrong terms, and Anjali subtracted too many times.
D. Anjali is correct because the polynomial is of degree 2.

Sagot :

GinnyAnsweridn
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.
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