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Chapter 11: Problem 11

Find the sum of each series. \(\sum_{i=1}^{10} 3\)

Short Answer

Expert verified
The sum is 30.

Step by step solution

01

Understand the Series

The series given is \(\text{sum of 3 from } i=1 \text{ to } i=10\): \(\sum_{i=1}^{10} 3\).
02

Identify the Number of Terms

In the series \(\sum_{i=1}^{10} 3\), we see that it starts from i=1 and goes up to i=10. This means there are 10 terms.
03

Calculate the Sum of the Series

In this series, each term is 3. Since there are 10 terms, the total sum is calculated as: \(3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3\) which is equal to \10 \times 3 = 30\.

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Key Concepts

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

series notation
Series notation is a way to represent the sum of a sequence of numbers formally and compactly. When we look at the series representation \(\sum_{i=1}^{10} 3\), it's telling us to add the number 3 exactly 10 times. The symbol `\(\sum\)` represents the summation, and the limits of summation (in this case, from i=1 to i=10) indicate the range of values that
arithmetic series
An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. In the given exercise, the series \(\sum_{i=1}^{10} 3\) can be seen as an arithmetic series where every term is the same. Here, each term in the series is 3, and there are 10 such terms. Arithmetic series have specific formulas to simplify their sum calculations, but in this case, since each term is the same, we can sum the terms directly with a simple multiplication.
sum calculation
Calculating the sum of the series \(\sum_{i=1}^{10} 3\) involves the following steps:
  • Step 1: Identify the number of terms. Here, i ranges from 1 to 10, giving us 10 terms.
  • Step 2: Recognize that each term in this series is a constant value of 3.
  • Step 3: Multiply the constant value (3) by the number of terms (10). This can be written as 3 \times\ 10 = 30.
Therefore, the sum of the series is 30. Breaking this down, you'll see that adding 3 together ten times (3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3) simply accumulates to 30. A simple yet fundamental example of series summation.

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Most popular questions from this chapter

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