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Questions and Answers
What is the general formula for the n-th term of an arithmetic sequence?
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?
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?
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?
If an investment doubles every 5 years, what is the common ratio of the geometric sequence representing its growth?
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In financial applications, which of the following scenarios best fits the model of a geometric series?
In financial applications, which of the following scenarios best fits the model of a geometric series?
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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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