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Questions and Answers

What is the general formula for the n-th term of an arithmetic sequence?

  • $a_n = r(n - 1) + a_1$
  • $a_n = a_1 r^{n-1}$
  • $a_n = n(d + a_1)$
  • $a_n = a_1 + (n - 1)d$ (correct)
  • Which of the following represents the sum of the first n terms of a geometric series?

  • $S_n = a(1 - r^n)/(1 - r)$ (correct)
  • $S_n = n(a + r)/2$
  • $S_n = ar^n - a$
  • $S_n = a + r(n - 1)$
  • What is the condition for an infinite geometric series to converge?

  • The first term must be greater than one.
  • The first term must be negative.
  • The common ratio must be equal to zero.
  • The common ratio must be less than one in absolute value. (correct)
  • If an investment doubles every 5 years, what is the common ratio of the geometric sequence representing its growth?

    <p>$2^{1/5}$</p> Signup and view all the answers

    In financial applications, which of the following scenarios best fits the model of a geometric series?

    <p>The appreciation of property value over time due to compounding interest.</p> Signup and view all the answers

    Study Notes

    Arithmetic Sequences

    • The general formula for the n-th term of an arithmetic sequence is an = a1 + (n - 1)d, where:
      • an is the n-th term
      • a1 is the first term
      • d is the common difference

    Geometric Series

    • The sum of the first n terms of a geometric series is represented by Sn = a(1 - rn) / (1 - r), where:
      • Sn is the sum of the first n terms
      • a is the first term
      • r is the common ratio

    Convergence of Infinite Geometric Series

    • An infinite geometric series converges if the absolute value of the common ratio, |r|, is less than 1 (|r| < 1).

    Investment Growth

    • If an investment doubles every 5 years, the common ratio of the geometric sequence representing its growth is 2. This is because the investment value is multiplied by 2 every 5 years.

    Geometric Series in Financial Applications

    • Compound interest best fits the model of a geometric series. In compound interest, the interest earned in each period is added to the principal, and the next period's interest is calculated on the new, larger principal. This creates a geometric sequence where the common ratio is 1 plus the interest rate.

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