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Magazine Chapter 9: Problem 17
In Exercises 17 - 28 , use the formulas in Equation 9.2 to find the sum. $$ \sum_{n=1}^{10} 5 n+3 $$
Short Answer
Expert verified
The sum of the series is 305.
Step by step solution
01
Expand the Series
The expression \( \sum_{n=1}^{10} (5n + 3) \) represents the sum of terms from \( n = 1 \) to \( n = 10 \). This is an arithmetic series where each term follows the pattern \( 5n + 3 \). List the first few terms up to \( n = 10 \). They are: \( 8, 13, 18, 23, 28, 33, 38, 43, 48, 53 \).
02
Identify the Series Type
Understand that the series can be split into two separate sums: one for the term \( 5n \) and another for the constant term \( 3 \). This follows from the property \( \sum_{n=1}^{N} (a_n + b) = \sum_{n=1}^{N} a_n + \sum_{n=1}^{N} b \).
03
Calculate the Sum of the 5n Part
For the part \( 5n \), factor the constant out of the summation: \( \sum_{n=1}^{10} 5n = 5 \sum_{n=1}^{10} n \). Use the formula for the sum of the first N natural numbers, \( \sum_{n=1}^{N} n = \frac{N(N+1)}{2} \), to find \( \sum_{n=1}^{10} n = \frac{10(10+1)}{2} = 55 \). Therefore, \( 5 \sum_{n=1}^{10} n = 5 \times 55 = 275 \).
04
Calculate the Sum of the Constant Part
Since \( 3 \) is a constant added to each term, the sum of \( \sum_{n=1}^{N} 3 \) is simply \( 3N \). For \( N = 10 \), we have \( 3 \times 10 = 30 \).
05
Add the Results from Earlier Steps
Add the results from Step 3 and Step 4 to get the total sum of the series. \( 275 + 30 = 305 \). The sum of the series \( \sum_{n=1}^{10} (5n + 3) \) is thus 305.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sum of Series
Summing a series involves adding a sequence of terms together. In our exercise, the series given is \( \sum_{n=1}^{10} (5n + 3) \). To calculate this, you undertake the process of individually identifying and adding each term in the series from \( n = 1 \) to \( n = 10 \).
In an arithmetic series like this one, there is a consistent amount added to each successive term, known as the common difference. To ensure accuracy, make sure all terms are correctly summed by following these steps:
In an arithmetic series like this one, there is a consistent amount added to each successive term, known as the common difference. To ensure accuracy, make sure all terms are correctly summed by following these steps:
- List all terms: like \( 8, 13, 18, \ldots \) all the way to \( 53 \).
- Add all terms together to get a single sum.
- Use techniques like factoring for efficiency, especially with large term numbers.
Arithmetic Progression
An arithmetic progression (AP) is a sequence where each term after the first is derived by adding a constant to the previous term. In our exercise, that constant, or common difference, is 5.
Here鈥檚 how you can clearly determine an arithmetic progression:
Here鈥檚 how you can clearly determine an arithmetic progression:
- Identify the first term, \( a_1 \). In this case, it is \( 8 \).
- Determine the common difference, \( d \), which is the change from one term to the next. For our series, \( d = 5 \).
- Each subsequent term can be obtained by the formula: \( a_{n} = a_1 + (n-1) \times d \).
Summation Formula
The summation formula for an arithmetic series simplifies evaluating the sum of a large number of terms by using a mathematical formula rather than manually adding each term.
For an arithmetic series, you can use:
\[S_N = \frac{N}{2} \times (a_1 + a_N)\]
Where:
\[S_N = \frac{N}{2} \times [2a_1 + (N-1) \times d]\]
These summation formulas are powerful tools for quickly finding the sum of terms in a series.
For an arithmetic series, you can use:
\[S_N = \frac{N}{2} \times (a_1 + a_N)\]
Where:
- \( S_N \) is the sum of the first \( N \) terms.
- \( a_1 \) is the first term.
- \( a_N \) is the last term, which is \( a_1 + (N-1) \times d \).
\[S_N = \frac{N}{2} \times [2a_1 + (N-1) \times d]\]
These summation formulas are powerful tools for quickly finding the sum of terms in a series.
Series Expansion
Series expansion involves writing a series in an expanded form by replacing the general formula with the specific terms that make up the series.
To perform series expansion:
To perform series expansion:
- Begin with the initial formula, like \( 5n + 3 \), and substitute \( n \) with 1, 2, 3, and so forth up to the last term.
- Write out each term explicitly: \( 5(1) + 3, 5(2) + 3, \ldots, 5(10) + 3 \).
- Calculate each term: leading to the series \( 8, 13, 18, \ldots, 53 \).
