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Series quiz

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  • What is the formula for the sequence discussed in the lesson?
    The formula is a_n = n^2 × (n-1)!. It combines a squared term and a factorial.
  • How do you find the first term of the sequence a_n = n^2 × (n-1)!?
    Plug n = 1 into the formula to get a_1 = 1^2 × 0! = 1 × 1 = 1.
  • What is the value of 0 factorial (0!)?
    0! is defined as 1. This is a standard mathematical convention.
  • How do you calculate the second term of the sequence?
    Plug n = 2 into the formula: a_2 = 2^2 × 1! = 4 × 1 = 4.
  • What is the value of 1 factorial (1!)?
    1! is equal to 1. It is the product of all positive integers up to 1.
  • How do you find the third term of the sequence?
    Plug n = 3 into the formula: a_3 = 3^2 × 2! = 9 × 2 = 18.
  • What is the value of 2 factorial (2!)?
    2! is equal to 2 × 1 = 2.
  • How do you calculate the fourth term of the sequence?
    Plug n = 4 into the formula: a_4 = 4^2 × 3! = 16 × 6 = 96.
  • What is the value of 3 factorial (3!)?
    3! is equal to 3 × 2 × 1 = 6.
  • What are the first four terms of the sequence a_n = n^2 × (n-1)!?
    The first four terms are 1, 4, 18, and 96.
  • Why is it acceptable to have a factorial in a sequence formula?
    Factorials can be part of sequence formulas and are treated like any other mathematical operation.
  • What operation do you perform first when evaluating a_n = n^2 × (n-1)!?
    First, calculate n^2, then compute (n-1)! and multiply the results.
  • How do you compute (n-1)! for n = 4?
    For n = 4, (n-1)! = 3! = 3 × 2 × 1 = 6.
  • What is the process for finding terms in a sequence with a factorial?
    Substitute the desired n value, compute the factorial, and multiply by the other part of the formula.
  • Does the sequence a_n = n^2 × (n-1)! continue beyond n = 4?
    Yes, the sequence continues for higher values of n using the same formula.