Problem 59 - Algebra for College Students

Question

$$ \sum_{i=1}^{45}(5 i+2) \quad 5265 $$

Step-by-Step Solution

Answer

The sum is 5265, hence the given total is correct.

Step 1: Understand the Problem

The exercise requires us to find the sum of the sequence \(5i + 2\) from \(i=1\) to \(i=45\), and to determine if it equals 5265.

Step 2: Write the General Expression for the Series

The series can be expressed as \(\sum_{i=1}^{45} (5i + 2)\). We need to find the sum of this series.

Step 3: Separate and Simplify the Series

Separate the expression \( \sum_{i=1}^{45} (5i + 2) = \sum_{i=1}^{45} 5i + \sum_{i=1}^{45} 2 \). This can be simplified to two separate sums: \(5 \sum_{i=1}^{45} i\) and \(2 \sum_{i=1}^{45} 1\).

Step 4: Calculate the First Part \(5 \sum_{i=1}^{45} i\)

Recall that the sum of the first \(n\) integers is \(\frac{n(n+1)}{2}\). Here, \(n=45\), so \( \sum_{i=1}^{45} i = \frac{45 \times 46}{2} = 1035\). Hence, \(5 \sum_{i=1}^{45} i = 5 \times 1035 = 5175\).

Step 5: Calculate the Second Part \(2 \sum_{i=1}^{45} 1\)

Since \( \sum_{i=1}^{45} 1 \) simply counts the numbers from 1 to 45, this sum is equal to 45. So, \(2 \times 45 = 90\).

Step 6: Combine the Results

Combine the results from Steps 4 and 5: \(5175 + 90 = 5265 \).

Step 7: Verify the Result

Compare the result from Step 6 with the provided total. Since both are \(5265\), the calculation confirms that the sum is correct.

Key Concepts

Arithmetic Series

An arithmetic series is a fascinating mathematical sequence where the difference between consecutive terms is constant. Such sequences have a predictable pattern, making them easier to sum. Let's consider an example: in the arithmetic sequence where each term increases by a fixed number, the sequence might look like this: 1, 3, 5, 7, \ldots\. The difference between each term is 2. This difference is the key to calculating the sum efficiently.The formula to find the sum of an arithmetic series is derived from the formula for the sum of the first \( n \) terms: \( S_n = \frac{n}{2} (a + l) \),where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. For our exercise, the arithmetic series is generated by the formula \( 5i + 2 \). This generates the terms, starting from \( 7 \) and increasing by 5 each time.Understanding this concept helps in recognizing the pattern and determining the sum quickly.

Summation Notation

Summation notation, often represented by the Greek letter \( \Sigma \), is a concise way of expressing the sum of a sequence of terms. In mathematics, the notation \( \sum_{i=1}^{n} \) means "sum the terms from \( i=1 \) to \( i=n \)". The terms to be summed are typically given by a formula that generates a sequence, such as \( 5i + 2 \) in our exercise.This notation is particularly powerful because it provides a clear and concise way to express series that can otherwise be cumbersome. Here, for example, instead of listing terms one by one, we use \( \sum_{i=1}^{45}(5i + 2) \) to represent the entire arithmetic series.This method allows mathematicians and students alike to focus on understanding the structure of sequences and series rather than getting bogged down with long calculations. With practice, interpreting summation notation becomes intuitive, making it an essential tool in mathematics.

Series Simplification

Series simplification is a valuable technique in mathematics, helping to break down complex expressions into manageable parts. This is particularly useful when summing series with multiple components.In our example, the expression \( \sum_{i=1}^{45} (5i + 2) \) was split into two simpler series: \( 5 \sum_{i=1}^{45} i \) and \( 2 \sum_{i=1}^{45} 1 \). Let's explore why this works:

• Split the Series: By separating into smaller sums, we make calculations easier. In our case, we calculated the sum of \( 5i \) and the sum of constants.
• Calculate Each Sum Individually: The arithmetic inside the series can be computed separately. This gives better clarity over each part’s contribution to the overall sum.
• Combine the Results: Once individual sums are calculated, they are added together to obtain the final sum, which is much simpler than working with the initial expression directly.

By employing series simplification, complicated sums become approachable, enabling more efficient problem-solving methods in mathematics.