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Sequences quiz

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  • What is a sequence in mathematics?
    A sequence is an ordered list of numbers, where each number is called a term or element of the sequence.
  • How do you determine if a sequence is finite or infinite?
    A sequence is finite if it has a definite end, and infinite if it continues indefinitely, often indicated by '...'.
  • What is the difference between a function and a sequence regarding their inputs?
    Functions can take any real number as input, while sequences only take positive integers as inputs, called indexes.
  • How do you find the nth term of a sequence using a general (explicit) formula?
    You substitute the value of n into the explicit formula to calculate the nth term directly.
  • What is a recursive formula for a sequence?
    A recursive formula defines each term based on one or more previous terms in the sequence, rather than directly using n.
  • How do you find the next term in a sequence using a recursive formula?
    You use the previous term(s) and apply the recursive rule to calculate the next term.
  • What pattern do arithmetic sequences follow?
    Arithmetic sequences have a constant difference between consecutive terms, such as adding the same number each time.
  • How can you recognize a sequence with alternating signs?
    A sequence with alternating signs often uses (-1) raised to the n or (n+1) power in its formula.
  • What does the formula a_n = n/(n+1) generate as a sequence?
    It generates a sequence where the numerator increases by 1 each time, and the denominator is always one more than the numerator.
  • How do you adjust a general formula if the first term does not match the index?
    You add, subtract, multiply, or divide by a constant to shift the formula so the first term matches the sequence.
  • What type of sequence is represented by a_n = 2^n?
    This is an exponential sequence, where each term is a power of 2.
  • Why might a recursive formula be preferred over an explicit formula?
    A recursive formula is often easier to use when you only need the next few terms and the explicit formula is hard to find.
  • How do you find the first three terms of the sequence a_n = n^2?
    Plug in n = 1, 2, and 3 to get 1, 4, and 9 as the first three terms.
  • What does the notation a_n-1 mean in a recursive formula?
    It refers to the previous term in the sequence, the term immediately before a_n.
  • How do you find the next term in the sequence defined by a_1 = 1 and a_n = n * a_n-1?
    Multiply the current index n by the previous term to get the next term; for example, a_2 = 2 * a_1 = 2.