Discuss With Class And Try To Solve 2-3 Problems Each Be Sur

Discuss With Class And Try To Solve 2 3 Problems Eachbe Sure To Solve

Discuss with class and try to solve 2-3 problems each Be sure to solve some of the Textbook application problems using two important formulas, the first one for the nth term of the sequence and the second one for the SUM of the first n terms of the sequence: Formula #1. an = a1 + (n-1)d, where a1 = the first term and d = common difference Formula #2. which is the formula for a1 + a2 + a3 + ... + an

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Arithmetic sequences and series are fundamental concepts in mathematics that describe patterns of numbers progressing in a specific manner. An arithmetic sequence is a list of numbers in which each term after the first is obtained by adding a constant difference to the previous term. Understanding these concepts allows us to analyze, model, and solve real-world problems effectively.

Understanding Arithmetic Sequences

An arithmetic sequence is defined by its first term, denoted as a₁, and its common difference, d. The sequence progresses by adding d to each term to generate the next. For example, consider the sequence 3, 7, 11, 15, ..., where the first term a₁ is 3, and the common difference d is 4. This sequence can be expressed using a general formula for any term in the sequence.

Formula for the nth Term

The nth term (a_n) of an arithmetic sequence can be expressed as:

an = a₁ + (n - 1)d

This formula computes the value of any term based on its position (n), given the first term and the common difference. It is crucial for solving problems involving specific terms within the sequence.

Arithmetic Series and Their Sum

An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first n terms, denoted as S_n, can be calculated using the formula:

S_n = (n/2) [2a₁ + (n - 1)d]

This formula is particularly useful for evaluating total amounts over a number of terms, such as cumulative salaries or total seats in a stadium segment.

Application Problems and Solutions

Problem 1: Company Salary Comparison After 10 Years

Company A pays an initial salary of $24,000 with yearly raises of $1,600. Company B pays $28,000 initially with yearly raises of $1,000. Which company will pay more in year 10, and by how much?

Using the nth term formula: a_n = a₁ + (n - 1)d

For Company A in year 10:

a₁₀ = 24,000 + (10 - 1) × 1,600 = 24,000 + 14,400 = $38,400

For Company B in year 10:

a₁₀ = 28,000 + (10 - 1) × 1,000 = 28,000 + 9,000 = $37,000

Therefore, in year 10, Company A pays $38,400, while Company B pays $37,000. Company A pays $1,400 more than Company B in year 10.

Problem 2: Total Salary Over 10 Years

Company A: starting at $23,000 with raises of $1,200 annually. Company B: starting at $26,000 with raises of $800 annually. Find which pays more over 10 years and the difference.

First, the total sum of salaries over 10 years for each company:

S_n = (n/2) [2a₁ + (n - 1)d]

For Company A:

S₁₀ = (10/2) [2×23,000 + (10 - 1)×1,200] = 5 [46,000 + 10,800] = 5 × 56,800 = $284,000

For Company B:

S₁₀ = (10/2) [2×26,000 + (10 - 1)×800] = 5 [52,000 + 7,200] = 5 × 59,200 = $296,000

Thus, over ten years, Company B pays a total of $296,000, which is $12,000 more than Company A.

Problem 3: Total Salary Over 10 Years with Different Starting Points and Raises

A company offers a starting salary of $33,000 with yearly raises of $2,500. Find total salary over ten years.

The formula:

S_n = (n/2) [2a₁ + (n - 1)d]

Applying the formula:

S₁₀ = (10/2) [2×33,000 + (10 - 1)×2,500] = 5 [66,000 + 22,500] = 5 × 88,500 = $442,500

Total salary over 10 years is $442,500.

Additional Problems and Applications

Problem 4: General Term for Specific Arithmetic Sequences

(a) Sequence with a₂ = 4 and a₆ = 16:

Using the formula a_n = a₁ + (n - 1)d, we find d and a₁.

From a₂: a₂ = a₁ + d = 4

From a₆: a₆ = a₁ + 5d = 16

Subtracting these equations:

(a₁ + 5d) - (a₁ + d) = 16 - 4

4d = 12 → d = 3

Plugging d back into a₂: a₁ + 3 = 4 → a₁ = 1

General term: a_n = 1 + (n - 1)×3 = 3n - 2

Problem 5: Sequence with Known Terms

(a) Sequence with a₃ = 7 and a₈ = 17:

Using the same method, find d:

a₃ = a₁ + 2d = 7

a₈ = a₁ + 7d = 17

Subtract equations:

(a₁ + 7d) - (a₁ + 2d) = 17 - 7

5d = 10 → d = 2

Plug d into a₃: a₁ + 2×2 = 7 → a₁ + 4 = 7 → a₁ = 3

General term: a_n = 3 + (n - 1)×2 = 2n + 1

Understanding Series versus Sequence

An arithmetic sequence is a list of numbers following a specific pattern, such as 2, 4, 6, 8, ... whereas an arithmetic series is the sum of the terms of the sequence. For example, the series of the first four terms: 2 + 4 + 6 + 8 = 20.

Sequences focus on the individual terms, while series concern the total sum of multiple terms, often calculated using the formula for the sum of an arithmetic series.

Conclusion

Mastering arithmetic sequences and series equips students with tools to model and analyze real-world patterns, from financial calculations to structural arrangements. By applying formulas for the nth term and the sum of terms, one can solve complex problems like salary progression, population growth, or resource allocation efficiently. These foundational concepts also pave the way for understanding more advanced mathematical topics like geometric sequences and calculus.

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