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? a) Shayna is correct. Jamal used the wrong terms, and Anjali subtracted too many times. b) Anjali is correct because the polynomial is a degree of 2. c) Anjali is correct. Jamal and Shayna both calculated 1st differences. d) Jamal is correct. Shayna calculated based on the wrong terms, and Anjali subtracted too many times.

1 answer

To find out who is correct regarding the second differences of the sequence given by the formula \( a_n = n^2 - 3 \), we need to start by calculating the first differences and then the second differences of the sequence.

  1. Calculate the 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 \( n = 5 \): \( a_5 = 5^2 - 3 = 25 - 3 = 22 \)

    Thus, the sequence is: \[ -2, 1, 6, 13, 22 \]

  2. Calculate the first differences: \[ \text{1st differences}: \]

    • \( 1 - (-2) = 3 \)
    • \( 6 - 1 = 5 \)
    • \( 13 - 6 = 7 \)
    • \( 22 - 13 = 9 \)

    So, the first differences are: \[ 3, 5, 7, 9 \]

  3. Calculate the second differences: \[ \text{2nd differences}: \]

    • \( 5 - 3 = 2 \)
    • \( 7 - 5 = 2 \)
    • \( 9 - 7 = 2 \)

    Thus, the second differences are: \[ 2, 2, 2 \]

Since all the second differences are the constant value of 2, Anjali is correct.

The final answer is: c) Anjali is correct. Jamal and Shayna both calculated 1st differences.