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

1 answer

To determine who is correct among Shayna, Jamal, and Anjali regarding the second differences of the sequence defined by the formula \( a_n = n^2 - 3 \), let's find the first and second differences step by step.

  1. Calculate the first few terms of the sequence:

    • \( a_1 = 1^2 - 3 = 1 - 3 = -2 \)
    • \( a_2 = 2^2 - 3 = 4 - 3 = 1 \)
    • \( a_3 = 3^2 - 3 = 9 - 3 = 6 \)
    • \( a_4 = 4^2 - 3 = 16 - 3 = 13 \)
    • \( a_5 = 5^2 - 3 = 25 - 3 = 22 \)

    The sequence is \( -2, 1, 6, 13, 22 \).

  2. Calculate the first differences:

    • \( a_2 - a_1 = 1 - (-2) = 3 \)
    • \( a_3 - a_2 = 6 - 1 = 5 \)
    • \( a_4 - a_3 = 13 - 6 = 7 \)
    • \( a_5 - a_4 = 22 - 13 = 9 \)

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

  3. Calculate the second differences:

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

    The second differences are all \( 2 \).

Now, based on our calculations:

  • Anjali stated that the second differences are \( 2 \), which is correct.
  • Shayna claimed the second differences are \( 5 \).
  • Jamal claimed the second differences are \( 7 \).

Conclusion:

Anjali is correct because the second differences are constant at \( 2 \). Thus, the response is:

Anjali is correct because the polynomial is a degree of 2.