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Arithmetic Sequences quiz

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이 집합의 용어 (15)

  • What is an arithmetic sequence?
    An arithmetic sequence is a sequence where each term differs from the previous one by a constant amount called the common difference.
  • How do you identify the common difference in an arithmetic sequence?
    Subtract any term from the next consecutive term; the result is the common difference d.
  • What is the formula for the nth term (a_n) of an arithmetic sequence?
    The formula is a_n = a_1 + (n-1)·d, where a_1 is the first term and d is the common difference.
  • If the first term of an arithmetic sequence is 2 and the common difference is 4, what is the fifth term?
    The fifth term is a_5 = 2 + (5-1)·4 = 18.
  • How do you find the common difference if you know the first two terms are 10 and 8?
    Subtract the first term from the second: 8 - 10 = -2, so the common difference is -2.
  • What does a negative common difference indicate about an arithmetic sequence?
    A negative common difference means each term is less than the previous one; the sequence decreases.
  • How do you find the next two terms of the sequence 10, 8, ...?
    Subtract the common difference (-2) from each term: 8 - 2 = 6, then 6 - 2 = 4.
  • What information do you need to write the general term of an arithmetic sequence?
    You need the first term (a_1) and the common difference (d).
  • How can you simplify the general term formula a_n = 2 + (n-1)·4?
    Distribute and combine like terms: a_n = 4n - 2.
  • How do you find the 20th term of an arithmetic sequence using the general term formula?
    Plug n = 20 into the formula: a_20 = a_1 + 19·d.
  • If a_1 = 8 and d = -6, what is the general term formula for the sequence?
    The general term is a_n = 8 + (n-1)·(-6).
  • If you know a_1 = 2 and a_5 = 14, how do you find the common difference?
    Set up the equation 14 = 2 + 4d, solve for d to get d = 3.
  • Why do you multiply the common difference by (n-1) in the general term formula?
    Because you add the common difference one less time than the term number to reach the nth term from the first term.
  • What is the general term formula for a sequence with a_1 = 2 and d = 3?
    The general term is a_n = 2 + (n-1)·3.
  • How can you use the general term formula to find any term in an arithmetic sequence without listing all terms?
    Plug the desired term number into the formula a_n = a_1 + (n-1)·d to calculate the value directly.