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Chapter 10: Problem 55

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

Short Answer

Expert verified
The sum is 36.

Step by step solution

01

Understand the Series Expression

The given series is \( \sum_{n=1}^{6}(2n-1) \). This means we need to evaluate the expression \( 2n-1 \) for each integer \( n \) from 1 to 6, and then sum all these values.
02

Evaluate the Terms Individually

Calculate each term of the series:- For \( n = 1 \), the term is \( 2(1)-1 = 1 \).- For \( n = 2 \), the term is \( 2(2)-1 = 3 \).- For \( n = 3 \), the term is \( 2(3)-1 = 5 \).- For \( n = 4 \), the term is \( 2(4)-1 = 7 \).- For \( n = 5 \), the term is \( 2(5)-1 = 9 \).- For \( n = 6 \), the term is \( 2(6)-1 = 11 \).
03

Sum the Evaluated Terms

Now, sum all the terms obtained:\[ 1 + 3 + 5 + 7 + 9 + 11 \]
04

Perform the Sum Calculation

Add the terms together:\[ 1 + 3 + 5 + 7 + 9 + 11 = 36 \]
05

Conclusion

The sum of the series \( \sum_{n=1}^{6}(2n-1) \) is 36.

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Key Concepts

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

Series Evaluation
When working with finite series, like \( \sum_{n=1}^{6}(2n-1) \), the goal is to evaluate the sum by adding up the values of a given expression for each term in the sequence. A finite series has a distinct number of terms. In this case, we calculate the expression \( (2n - 1) \) from \( n = 1 \) to \( n = 6 \).
To do this:
  • Substitute each integer from 1 to 6 into the expression \( 2n-1 \).
  • Calculate each term's value.
  • Add all the resulting numbers together to get the sum.
Understanding this systematic approach to evaluating a series is crucial in mathematics.Breaking down each step ensures accuracy, especially in sets with many terms. While simple series like the one here may seem straightforward, the logical clarity gained through this exercise becomes invaluable for more complex problems.
Sum of Arithmetic Sequence
An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. The series \( \sum_{n=1}^{6}(2n-1) \) represents such an arithmetic sequence where each number is formed by a regular pattern.
To find the sum of an arithmetic sequence, use the formula:\[ S_n = \frac{n}{2} \times (a + l) \]where:
  • \( n \) is the number of terms,
  • \( a \) is the first term,
  • \( l \) is the last term.
For \( n = 6 \) here with terms starting at 1 and ending at 11:
  • \( a = 1 \)
  • \( l = 11 \)
Finally, plugging into the formula results in \[ S_6 = \frac{6}{2} \times (1 + 11) = 3 \times 12 = 36 \].
This reveals the sum. Recognizing the arithmetic nature is useful, as it streamlines calculations, especially with bigger sets.
Precalculus Problem Solving
Precalculus provides the fundamental skills required to solve a wide array of mathematical problems. It introduces concepts such as arithmetic sequences and series evaluation. Tackling problems like \( \sum_{n=1}^{6}(2n-1) \) prepares students for more advanced calculus and analytical techniques.
Improving precalculus problem solving involves:
  • Understanding the core principles behind each type of sequence and series.
  • Practicing the systematic breakdown of problems into smaller, manageable parts.
  • Applying formulas, such as the one for arithmetic sequence sums, confidently.
  • Checking calculations for accuracy to solidify understanding.
These practices not only improve mathematical reasoning but also encourage precision in calculations. They set the stage for tackling complex equations and functions in future math courses.

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Most popular questions from this chapter

Evaluate each finite series. $$\sum_{n=0}^{3}(-x)^{n}$$

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