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Chapter 10: Problem 66

Evaluate the sum. For each sum, state whether it is arithmetic or geometric. Depending on your answer, state the value of d or \(r\). $$\sum_{k=0}^{6}(2(2 k+4)-2(k+1))$$

Short Answer

Expert verified
The given series is arithmetic, with a common difference (d) of 2. The sum of the series is 84.

Step by step solution

01

Simplify Individual Terms

To evaluate the series, start by simplifying the expression given for each term. Term k in the series is described by \(2(2k+4)-2(k+1)\). Simplifying this gives \(4k + 8 - 2k - 2\), which further simplifies to \(2k + 6\).
02

Calculate First Few Terms

Calculate the first few terms of the series to help identify if it's arithmetic or geometric. Use the formula obtained in the previous step: first term (k=0) is 6, the second term (k=1) is 8, and the third term (k=2) is 10.
03

Identify the Type of Series

The differences between the terms calculated in the previous step are 2, which is consistent. Therefore, the series is an arithmetic series.
04

Determine the Common Difference

The common difference (d) in an arithmetic series is the difference between two adjacent terms. In this case, d is 2, as observed from the terms calculated in step 2.
05

Evaluate the Sum

To calculate the sum of an arithmetic series, use the formula \[n/2 * (first\_term + last\_term)\] where \(n\) is the number of terms, which is 7 in this case. The first term is 6 and the last term \((2(6)+6)\) is 18. So the sum is \[7/2 * (6 + 18) = 84\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Common Difference
In the context of arithmetic sequences, the common difference is a crucial concept. It tells us how much we need to add (or subtract) to move from one term to the next in the sequence. Identifying the common difference allows us to determine if a series is arithmetic. In an arithmetic series, every number in the sequence increases or decreases by a fixed amount, which is the common difference.

  • To find the common difference, subtract any term from the subsequent term.
  • For this exercise, subtract the first term (6) from the second term (8) to find the common difference of 2.
  • This difference (denoted as 'd') remains constant throughout the entire arithmetic sequence.
Understanding the common difference is essential as it allows us to generate all other terms in the sequence and facilitates calculating sums in later steps.
Basics of an Arithmetic Sequence
An arithmetic sequence, also known as an arithmetic progression, is a list of numbers where each term is obtained by adding a constant value to its predecessor. This characteristic makes arithmetic sequences predictable and easy to work with. In our exercise, we identified the series as arithmetic because each subsequent term increased by the common difference of 2.

Each term of the sequence can be expressed using the formula:
  • \( a_n = a_1 + (n-1) imes d \)
  • Where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
In practical terms, knowing one term and the common difference lets us calculate any other term in the sequence. For example, knowing that our first term is 6, with a common difference of 2, we can ascertain that the sequence begins as 6, 8, 10, and so on.
Using the Series Sum Formula
The series sum formula for an arithmetic series is a handy tool for quickly adding up a sequence of numbers without summing each term individually. This formula is mathematically represented as:
  • \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
  • Where \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, and \( a_n \) is the last term in the sequence.
In our exercise, to find the total of the sequence from 6 to 18 (inclusive), we applied this formula:
  • We calculated \( n \), the total number of terms, which is 7.
  • The first term \( a_1 \) is 6, and the last term \( a_n \) is 18.
  • Using the formula \( S_n = \frac{7}{2} \times (6 + 18) = 84 \), the sum of this arithmetic series is 84.
Understanding the sum formula not only simplifies calculations but also enhances your ability to quickly evaluate other arithmetic series encountered in math.

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Most popular questions from this chapter

In this set of exercises, you will use sequences to study real-world problems. Investment An income-producing investment valued at \(\$ 2000\) pays interest at an annual rate of \(6 \% .\) Assume that the interest is taken out as income and therefore is not compounded. (a) Make a table in which you list the initial investment along with the total value of the investment-related assets (initial investment plus total interest earned) at the end of each of the first 4 years. (b) What is the total value of the investment-related assets after \(n\) years?

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