Question

Evaluate each series. $$\sum_{i=1}^{5}(2 i+1)$$

   Evaluate each series.
$$\sum_{i=1}^{5}(2 i+1)$$

Precalculus

Margaret L. Lial,… 6th Edition

Chapter 11, Problem 37

Instant Answer

Solved by Expert Ankit Gupta

07/20/2020

Step 1

When $i=1$, $2i+1=2(1)+1=3$. When $i=2$, $2i+1=2(2)+1=5$. When $i=3$, $2i+1=2(3)+1=7$. When $i=4$, $2i+1=2(4)+1=9$. When $i=5$, $2i+1=2(5)+1=11$. Now, let's add up these values: $3+5+7+9+11=35$. Show more…

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Evaluate each series. $$\sum_{i=1}^{5}(2 i+1)$$

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Key Concepts

Summation Notation

Summation notation is a concise way to represent the addition of a sequence of terms. It uses the sigma symbol (?) to indicate the sum of a sequence, with an index variable that takes on a set of consecutive values. This notation simplifies the expression of series and helps in generalizing problems in calculus and discrete mathematics.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term is generated by adding a constant difference to the previous term. Understanding arithmetic series is essential in many areas of mathematics, as it provides a foundation for analyzing linear patterns and solving problems involving regularly increasing or decreasing sequences.

Linearity of Summation

The linearity of summation is a fundamental property that allows the sum of a linear combination of terms to be separated into individual sums. This principle simplifies the evaluation of series by permitting the distribution of the summation operation over addition and the factoring out of constant coefficients, thereby facilitating more straightforward computations.

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