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Chapter 11: Problem 30

Evaluate each series. $$\sum_{i=1}^{6}(3 i-2)$$

Short Answer

Expert verified
The sum is 51.

Step by step solution

01

- Understand the Series

The given series is \(\sum_{i=1}^{6}(3i-2)\). This means you need to find the sum of the expression \(3i-2\) from \(i=1\) to \(i=6\).
02

- Write Out the Terms

Substitute the values from \(i=1\) to \(i=6\) into the expression \(3i-2\): \( (3(1)-2), (3(2)-2), (3(3)-2), (3(4)-2), (3(5)-2), (3(6)-2) \).
03

- Calculate Each Term

Calculate each term individually: \(3(1)-2 = 1, 3(2)-2 = 4, 3(3)-2 = 7, 3(4)-2 = 10, 3(5)-2 = 13, 3(6)-2 = 16 \).
04

- Add Up the Terms

Sum the calculated terms: \(1 + 4 + 7 + 10 + 13 + 16 \).
05

- Find the Total

Compute the total: \(1 + 4 + 7 + 10 + 13 + 16 = 51 \).

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Key Concepts

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

Arithmetic Series
An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. This difference is known as the common difference. In our exercise, the expression given is \(3i-2\), and it changes linearly with each increment of i.
To illustrate, let鈥檚 break down the provided example: \(3(1)-2 = 1\), \(3(2)-2 = 4\), \(3(3)-2 = 7\), and so on. Notice how there is a constant difference of 3 between consecutive terms (4 - 1, 7 - 4, etc.).
So, in an arithmetic series:
  • Each term increases (or decreases) by a fixed amount, called the common difference (d).
  • This pattern continues uniformly to form the entire series.
Understanding the concept of an arithmetic series is vital for tackling problems that require summation of terms with a uniform interval.
Finite Series
A finite series is a series that has a limited number of terms. In the given exercise, we are dealing with a finite series because we have to find the sum from \(i=1\) to \(i=6\). This automatically limits the series to 6 terms.
Finite series are easier to evaluate because they do not go on indefinitely. To evaluate a finite series, follow these steps:
  • Identify the expression or sequence to be summed.
  • Determine the range of terms (e.g., from 1 to 6).
  • Compute each term in the given range.
  • Sum up all these computed terms.
In the example provided, the series is \(\sum_{i=1}^{6}(3i-2)\), giving us a clear start and end point (from 1 to 6). Evaluating finite series often results in a specific numeric answer, in this case, 51.
Series Evaluation
Series evaluation involves calculating the sum of a sequence of terms systematically. To evaluate the series given in the problem \(\sum_{i=1}^{6}(3i-2)\), you can follow a step-by-step approach.

  • Start by understanding the series: Identify the general formula or rule of the series, in this case, \(3i-2\).
  • Next, write out the terms: Substitute each value of \(i\) from the range 1 to 6 into the formula to get the individual terms: \(1, 4, 7, 10, 13, 16\).
  • Calculate each term individually to ensure accuracy.
  • Finally, sum these terms: Add up all the terms \(1 + 4 + 7 + 10 + 13 + 16\).
This approach simplifies the problem by breaking it down into manageable steps. It is a reliable method to tackle similar summation problems accurately and systematically.

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Most popular questions from this chapter

Work each problem.One game in a state lottery requires you to pick 1 heart, 1 club, 1 diamond, and 1 spade, in that order, from the 13 cards in each suit. What is the probability of getting all four picks correct and winning \(\$ 5000 ?\)

Certain medical conditions are treated with a fixed dose of a drug administered at regular intervals. Suppose a person is given 2 mg of a drug each day and that during each 24 -hr period, the body utilizes \(40 \%\) of the amount of drug that was present at the beginning of the period. (a) Show that the amount of the drug present in the body at the end of \(n\) days is $$\sum_{i=1}^{n} 2(0.6)^{i}$$ (b) What will be the approximate quantity of the drug in the body at the end of each day after the treatment has been administered for a long period of time?

In how many ways can 5 players be assigned to the 5 positions on a basketball team, assuming that any player can play any position? In how many ways can 10 players be assigned to the 5 positions?

A city council is composed of 5 liberals and 4 conservatives. Three members are to be selected randomly as delegates to an urban convention. (a) How many delegations are possible? (b) How many delegations could have all liberals? (c) How many delegations could have 2 liberals and 1 conservative? (d) If 1 member of the council serves as mayor, how many delegations are possible that include the mayor?

Assume that \(n\) is a positive integer. Use mathematical induction to prove each statement S by following these steps. See Example \(I\). (a) Verify the statement for \(n=1\) (b) Write the statement for \(n=k\) (c) Write the statement for \(n=k+1\) (d) Assume the statement is true for \(n=k\). Use algebra to change the statement in part (b) to the statement in part (c). (e) Write a conclusion based on Steps (a)-(d). $$5+10+15+\dots+5 n=\frac{5 n(n+1)}{2}$$
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