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Geometric Sequences quiz

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  • What is the main difference between arithmetic and geometric sequences?
    Arithmetic sequences have a constant difference between terms, while geometric sequences have a constant ratio between terms.
  • What is the common ratio in a geometric sequence?
    The common ratio is the fixed number each term is multiplied by to get the next term in a geometric sequence.
  • How do you find the common ratio in a geometric sequence?
    Divide any term in the sequence by the previous term to find the common ratio.
  • What is the recursive formula for a geometric sequence?
    The recursive formula is a_n = a_(n-1) * r, where r is the common ratio.
  • Why must you specify the first term when writing a recursive formula for a geometric sequence?
    Because the recursive formula needs a starting value to generate the rest of the sequence.
  • How does the growth of geometric sequences compare to arithmetic sequences?
    Geometric sequences grow much faster than arithmetic sequences because they are exponential.
  • What is the general formula for the nth term of a geometric sequence?
    The general formula is a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.
  • What two pieces of information do you need to write the general formula for a geometric sequence?
    You need the first term (a_1) and the common ratio (r).
  • How do you use the general formula to find a specific term in a geometric sequence?
    Plug the desired term number (n) into the formula a_n = a_1 * r^(n-1).
  • Why is the general formula more useful than the recursive formula for finding high-index terms?
    Because the general formula allows you to calculate any term directly without computing all previous terms.
  • If a geometric sequence starts with 5 and has a common ratio of 4, what is the recursive formula?
    The recursive formula is a_n = a_(n-1) * 4 with a_1 = 5.
  • If a geometric sequence starts with 3 and has a common ratio of 2, what is the general formula?
    The general formula is a_n = 3 * 2^(n-1).
  • How do you calculate the 12th term of the sequence 5, 20, 80, 320,...?
    Use the formula a_12 = 5 * 4^(12-1) = 5 * 4^11 = 5 * 4194304 = 20,971,520.
  • What operation is used to get from one term to the next in a geometric sequence?
    Multiplication by the common ratio is used to get from one term to the next.
  • What is the effect of the operation used in geometric sequences on the sequence's growth?
    Multiplication causes geometric sequences to grow exponentially, leading to much larger terms compared to arithmetic sequences.