Book cover for Calculus with Applications

Calculus with Applications

Margaret L. Lial • Raymond N. Greenwell • Nathan P. Ritchey

ISBN #9781292108971

11th Edition

3,612 Questions

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224,424 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces sequences as functions defined on natural numbers and focuses on geometric sequences, where each term is produced by multiplying the previous term by a common ratio. It details how to calculate specific terms and the sum of a sequence using the formulas a? = a · r^(n-1) and S? = a( r? - 1 )/(r - 1). Practical applications, including depreciation of machines and annuities, are discussed to illustrate these concepts' real-world significance.

Learning Objectives

1

Explain the concept of a sequence as a function with a domain of natural numbers.

2

Differentiate between arithmetic and geometric sequences and identify their key properties.

3

Apply formulas to find specific terms and sums of geometric sequences.

4

Analyze practical applications including depreciation and annuities using geometric series concepts.

Key Concepts

CONCEPT

DEFINITION

Sequence

A function whose domain is a subset of the natural numbers, where each input corresponds to a term in the list.

Term (aₙ)

The nth element of a sequence; for example, aₙ = 2n indicates that the sequence's nth term is 2 multiplied by n.

Geometric Sequence

A sequence in which each term after the first is found by multiplying the preceding term by a constant called the common ratio (r).

Common Ratio (r)

The constant factor between consecutive terms in a geometric sequence, computed as r = aₙ₊₁ / aₙ.

Sum of a Geometric Sequence (Sₙ)

The sum of the first n terms of a geometric sequence; calculated for r ≠ 1 by the formula Sₙ = a( rⁿ - 1 ) / (r - 1).

Annuity

A series of equal payments made at regular intervals where each payment is compounded at a given interest rate.

Example Problems

Example 1

Exercises 1 - 6 List the first $n$ terms of the geometric sequence satisfying the following conditions. $$a_{1}=2, r=2, n=5$$

Example 2

Exercises 1 - 6 List the first $n$ terms of the geometric sequence satisfying the following conditions. $$a_{1}=3, r=2, n=5$$

Example 3

Exercises 1 - 6 List the first $n$ terms of the geometric sequence satisfying the following conditions. $$a_{1}=1 / 2, r=4, n=4$$

Example 4

Exercises 1 - 6 List the first $n$ terms of the geometric sequence satisfying the following conditions. $$a_{1}=2 / 3, r=6, n=3$$

Example 5

Exercises 1 - 6 List the first $n$ terms of the geometric sequence satisfying the following conditions. $$a_{3}=2, a_{4}=4, n=5$$
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Step-by-Step Explanations

QUESTION

Find the first four terms of the sequence defined by aₙ = -4n + 2.

STEP-BY-STEP ANSWER:

Step 1: Substitute n = 1: a₁ = -4(1) + 2 = -4 + 2 = -2.
Step 2: Substitute n = 2: a₂ = -4(2) + 2 = -8 + 2 = -6.
Step 3: Substitute n = 3: a₃ = -4(3) + 2 = -12 + 2 = -10.
Step 4: Substitute n = 4: a₄ = -4(4) + 2 = -16 + 2 = -14.
Final Answer: The first four terms are -2, -6, -10, and -14.

Finding a Term of a Sequence

QUESTION

Given the geometric sequence 6, 24, 96, 384, … find the 7th term.

STEP-BY-STEP ANSWER:

Step 1: Identify the first term a = 6 and compute the common ratio r = 24/6 = 4.
Step 2: Use the formula for the nth term: aₙ = a * r^(n-1).
Step 3: Substitute n = 7: a₇ = 6 * 4^(7-1) = 6 * 4^6.
Step 4: Calculate 4^6 = 4096, then a₇ = 6 * 4096 = 24576.
Final Answer: The 7th term of the sequence is 24,576.

Finding a Term in a Geometric Sequence

QUESTION

Find the sum of the first six terms of the geometric sequence 3, 12, 48, …

STEP-BY-STEP ANSWER:

Step 1: Identify the first term a = 3 and the common ratio r = 12/3 = 4.
Step 2: Use the sum formula Sₙ = a * (rⁿ - 1) / (r - 1) with n = 6.
Step 3: Compute r⁶: 4^6 = 4096.
Step 4: Substitute values: S₆ = 3 * (4096 - 1) / (4 - 1) = 3 * 4095 / 3.
Step 5: Simplify: S₆ = 4095.
Final Answer: The sum of the first six terms is 4095.

Finding the Sum of a Geometric Sequence

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Common Mistakes

  • Confusing the variable n as x; remember that n represents the term number in the sequence.
  • Mixing up arithmetic sequence formulas with geometric sequence formulas.
  • Forgetting that for a geometric sequence the exponent in the nth term is (n-1).
  • Not checking the condition r = 1 when using the sum formula, which requires a different approach.
  • Errors in exponentiation calculation, especially for large exponents.