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Chapter 12: Problem 32

Write out each series and evaluate it. $$ \sum_{i=1}^{6}(i+9) $$

Short Answer

Expert verified
75

Step by step solution

01

- Write out the series

Expand the sigma notation by substituting the values of i from 1 to 6 into the expression \(i + 9\). This gives: \( (1+9) + (2+9) + (3+9) + (4+9) + (5+9) + (6+9) \) .
02

- Simplify each term inside the series

Simplify each term individually: \(1+9=10, 2+9=11, 3+9=12, 4+9=13, 5+9=14, 6+9=15\). This results in: \(10 + 11 + 12 + 13 + 14 + 15\).
03

- Add the simplified terms

Add the simplified terms together: \(10 + 11 + 12 + 13 + 14 + 15 = 75\).

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Key Concepts

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

Evaluating Series
In algebra, evaluating series is a potent tool used to sum a sequence of numbers. In our exercise, we need to find the sum of the series using sigma notation. Sigma notation, represented by the Greek letter \(\sum\), is a concise way to denote the sum of a sequence. Here, we consider the series \(\sum_{i=1}^{6}(i+9)\). To evaluate this series, we first expand it by substituting the values of i from 1 to 6 into the expression \(i + 9\). This gives us \( (1+9) + (2+9) + (3+9) + (4+9) + (5+9) + (6+9)\). Breaking down the series step-by-step helps us systematically find the sum. It is essential to carefully track the terms to avoid mistakes.
Simplifying Terms
Simplifying terms is a crucial step in resolving series and simplifying mathematical expressions in general. After expanding our series \( (1+9) + (2+9) + (3+9) + (4+9) + (5+9) + (6+9)\), the next step is to simplify each term. Each addition inside the parentheses should be done individually: \( 1+9 = 10 \), \( 2+9 = 11 \), \( 3+9 = 12 \), \( 4+9 = 13 \), \( 5+9 = 14 \), and \( 6+9 = 15\). After simplification, we obtain the sequence \(10, 11, 12, 13, 14, 15\). Simplification reduces complexity and makes the series easier to sum in the final step.
Summation
Summation, or adding together a series of numbers, is the final step in evaluating the series. Once each term has been simplified, we combine them to find the total sum. From our previous steps, we have the simplified terms \(10 + 11 + 12 + 13 + 14 + 15\). Adding these together, we carry out the operation step-by-step: \(10 + 11 = 21\), \(21 + 12 = 33\), \(33 + 13 = 46\), \(46 + 14 = 60\), and finally \(60 + 15 = 75\). Therefore, the sum of the series \(\sum_{i=1}^{6}(i+9)\) is 75. Summation involves careful addition of each term to ensure accuracy in the final result.

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