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Chapter 10: Problem 52

Evaluate each finite series. $$\sum_{n=1}^{5} 7$$

Short Answer

Expert verified
The sum of the series is 35.

Step by step solution

01

Understand the Series

The series given is \( \sum_{n=1}^{5} 7 \). This means that we are summing the constant number 7 for each value of \( n \) from 1 to 5.
02

Count the Number of Terms

Determine how many terms we have to add. Since the index \( n \) runs from 1 to 5 in the series, we need to sum a total of 5 terms.
03

Calculate the Sum

Since each term in the series is 7 and there are 5 terms, multiply 7 by the number of terms: \[ 7 \times 5 = 35\].
04

Verify the Result

Recheck the calculation to ensure no errors were made:- There are 5 terms in total.- Each term is 7.- The sum is \( 7 \times 5 = 35 \). The result is correct.

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Key Concepts

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

Understanding Constant Series
In mathematics, a constant series is a type of finite series where each term in the sequence is the same. This means that you will see the same number repeated over and over again in the series. For example, in the series given by \( \sum_{n=1}^{5} 7 \), the number 7 is repeated for each of the 5 terms.

The beauty of a constant series lies in its simplicity. Since all terms are the same, calculating the sum is straightforward. Simply multiply the constant by the number of terms.

The key aspects of a constant series include:
  • All terms are identical.
  • Easy computation through multiplication.
  • No variations or changes in terms.
In practice, constant series calculations help in quick problem solving where uniformity across elements is expected.
Decoding Summation Notation
Summation notation is a compact way to represent the sum of a series of numbers. It's represented by the Greek letter sigma (\( \Sigma \)) and provides an efficient way to write long sums.

In the example \( \sum_{n=1}^{5} 7 \), the notation tells us:
  • The index of summation, \( n \), starts at 1 and ends at 5, meaning there are 5 terms.
  • The number being summed is 7, which remains constant.
Understanding summation notation requires attention to the index limits and the expression being summed. Summation reduces the visual clutter of writing out repeated addition and provides clarity in more complex problems.

Mastering Arithmetic Operations
Arithmetic operations form the basis for solving series problems. When dealing with constant series, the main arithmetic operation is multiplication. This is because adding the same number repeatedly can be quickly done by multiplying it by the count of terms in the series.

In our case, the series \( \sum_{n=1}^{5} 7 \) simplified further becomes:
  • Compute the total number of terms: 5 from \( n=1 \) to \( n=5 \).
  • Multiply the constant term 7 by the number of terms: \( 7 \times 5 \).
  • The result is 35.
Rechecking your calculations ensures accuracy, especially when dealing with larger series. Proper use of arithmetic operations aids in efficient and error-free evaluations.

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Most popular questions from this chapter

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