Problem 17 - Precalculus

Question

In Exercises 17 - 28 , use the formulas in Equation 9.2 to find the sum. $$ \sum_{n=1}^{10} 5 n+3 $$

Step-by-Step Solution

Answer

The sum of the series is 305.

Step 1: 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: \( 8, 13, 18, 23, 28, 33, 38, 43, 48, 53 \).

Step 2: 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 \( \sum_{n=1}^{N} (a_n + b) = \sum_{n=1}^{N} a_n + \sum_{n=1}^{N} b \).

Step 3: 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, \( 5 \sum_{n=1}^{10} n = 5 \times 55 = 275 \).

Step 4: 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 \( 3 \times 10 = 30 \).

Step 5: 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.

Key Concepts

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:

• 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.

Understanding sum of series formulas and calculations is crucial for evaluating these efficiently.

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’s 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: \( a_{n} = a_1 + (n-1) \times d \).

This logical structure of finding terms helps in systematically computing or verifying each term in a sequence.

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:

• \( S_N \) is the sum of the first \( N \) terms.
• \( a_1 \) is the first term.
• \( a_N \) is the last term, which is \( a_1 + (N-1) \times d \).

Alternatively, if the number of terms \( n \) and common difference \( d \) are known, you can also express the formula via:
\[S_N = \frac{N}{2} \times [2a_1 + (N-1) \times d]\]
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:

• 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 \( 8, 13, 18, \ldots, 53 \).

By expanding the series, you can visually interpret each term and validate your calculation, effectively ensuring the proper sum is obtained. This process builds intuition about sequences in series.