Ma1310 Week 9 Sequences And Notation Lab

Ma1310 Week 9 Sequences And Notationsthis Lab Requires You To Find

Describe an arithmetic sequence in two sentences.

Describe a geometric sequence in two sentences.

The sequence shown below is defined using a recursion formula. Write the first four terms of the sequence. a1 = 13 and an = an-1 + 8 for n ≥ 2.

A company offers a starting yearly salary of $33,000 with a raise of $2,500 per year. Find the total salary over a ten-year period.

Suppose you save $1 the first day of a month, $5 the second day, $25 the third day, and so on. That is, each day you save five times as much as you did the day before. What will you put aside for savings on the eighth day of the month?

Paper For Above instruction

Sequences are fundamental concepts in mathematics, representing ordered collections of numbers following specific rules. Understanding the characteristics of arithmetic and geometric sequences is essential for various applications, including finance, computer science, and natural sciences, as they model real-world phenomena involving consistent patterns over time.

An arithmetic sequence is characterized by a constant difference between consecutive terms. Specifically, if each term is obtained by adding a fixed amount to the previous term, the sequence exhibits a linear pattern. For example, the sequence 3, 7, 11, 15, 19, ... is arithmetic with a common difference of 4. This sequence can be succinctly described using the explicit formula for the nth term, an = a1 + (n - 1)d, where a1 is the first term, and d is the common difference. Arithmetic sequences are useful for modeling scenarios like steady savings or uniform payment plans.

A geometric sequence involves each term being obtained by multiplying the previous term by a fixed ratio. These sequences exhibit exponential growth or decay. For instance, the sequence 2, 4, 8, 16, 32, ... is geometric with a common ratio of 2. The general term of a geometric sequence can be expressed as an = a1 * rn-1, where a1 is the initial term and r is the common ratio. Geometric sequences are prevalent in modeling processes such as population growth, radioactive decay, and investment returns.

In solving problems involving sequences, recursive formulas often define terms based on previous terms. For example, the sequence defined by a1 = 13 and an = an-1 + 8 for n ≥ 2 is an arithmetic sequence with initial term 13 and common difference 8. To find the first four terms, we iteratively apply the recursive rule:

Starting with a1 = 13:

  • Term 1: 13
  • Term 2: 13 + 8 = 21
  • Term 3: 21 + 8 = 29
  • Term 4: 29 + 8 = 37

Hence, the first four terms are: 13, 21, 29, 37.

Financial applications can be modeled with arithmetic sequences, such as determining salary growth over years. For example, a starting salary of $33,000 with an annual raise of $2,500 can be represented as an arithmetic sequence with the first term a1 = 33,000 and common difference d = 2,500.

The total salary over ten years involves summing the first ten terms of this sequence. The formula for the sum of the first n terms of an arithmetic sequence is:

Sn = (n/2) * [2a1 + (n - 1)d]

Substituting the given values:

S10 = (10/2)  [233,000 + (10 - 1)2,500] = 5  [66,000 + 22,500] = 5 * 88,500 = $442,500

This accumulated salary illustrates how regular increases can significantly impact total earnings over time.

Further, the geometric progression can model savings plans where the amount saved each day increases exponentially. If you start saving $1 on the first day, and double your savings each subsequent day, the amount saved on day 8 can be determined using the general term of a geometric sequence:

an = a1 * rn - 1

Here, a1 = 1 and r = 2. Substituting the values to find the eighth day's savings:

a8 = 1 * 28 - 1 = 27 = 128

Thus, on the eighth day, you will set aside $128. This exponential savings pattern demonstrates how small recurring investments can grow substantially over a short period, emphasizing the power of compound growth in savings.

In conclusion, understanding sequences and their notation is invaluable for modeling and solving real-world problems involving growth, decay, and accumulation. Recognizing the differences between arithmetic and geometric sequences and being able to derive terms, sums, and ratios enable better analysis and decision-making in various domains.

References

  • Bittinger, M., Ellenbogen, J., & Quincy, D. (2017). Elementary and Intermediate Algebra. Pearson.
  • Larson, R., & Edwards, B. H. (2018). Precalculus with Limits. Cengage Learning.
  • Swokowski, E. W., & Cole, J. A. (2018). Algebra and Trigonometry. Cengage Learning.
  • Anton, H., Bivens, I., & Davis, S. (2014). Calculus: Early Transcendentals. Wiley.
  • Velleman, D. J. (2018). How to Prove It: A Structured Approach. Cambridge University Press.
  • Zeitz, G. J. (2019). The Art of Problem Solving, Volume 1: The Basics. Wiley.
  • Ross, S. (2018). Introduction to Probability and Statistics. Academic Press.
  • Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
  • Rosen, K. H. (2018). Discrete Mathematics and Its Applications. McGraw-Hill Education.
  • Lay, D. C. (2020). Linear Algebra and Its Applications. Pearson.

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