Problem 40 - Algebra 2

Question

Find the sum of each arithmetic series. $$ \sum_{k=7}^{11}(42-9 k) $$

Step-by-Step Solution

Answer

The sum is -195.

Step 1: Identify First and Last Terms

First, we identify the first and last terms of the series when substituted by the initial and final values of \(k\). For \(k = 7\), the first term is \(a_1 = 42 - 9 \times 7 = -21\). For \(k = 11\), the last term is \(a_n = 42 - 9 \times 11 = -57\).

Step 2: Calculate Number of Terms

Use the formula for the number of terms in the sequence: \(n = (11 - 7) + 1 = 5\). So there are 5 terms in the series.

Step 3: Determine Common Difference

The common difference \(d\) is the difference between consecutive terms. When \(k=7\), we have \(a_2 = 42 - 9 \times 8 = -30\). So \(d = a_2 - a_1 = -30 - (-21) = -9\).

Step 4: Verify Arithmetic Series

Confirm the series is arithmetic by checking that the difference is consistent; it matches the general formula for any two consecutive terms: \( (a_{k+1} - a_k) = -9 \).

Step 5: Use Formula for Sum of Arithmetic Series

The sum of an arithmetic series can be calculated as \( S_n = \frac{n}{2} (a_1 + a_n) \). Substitute the known values: \( S_5 = \frac{5}{2}(-21 + -57) = \frac{5}{2}(-78) = -195 \).

Key Concepts

First and Last Terms

In any arithmetic series, the first and last terms are crucial starting points. These terms define the sequence and are used to calculate other series properties like the sum. To find the first term, simply substitute the initial value of the variable given in the series equation. For instance, in the example:

• The first term is calculated by substituting the smallest value of \( k \) into the expression \( 42 - 9k \).
• For \( k = 7 \), this gives us \( a_1 = 42 - 9 \times 7 = -21 \).

Similarly, for the last term, substitute the final value of the variable. For \( k = 11 \), the last term is:

• \( a_n = 42 - 9 \times 11 = -57 \).

Number of Terms

The number of terms in an arithmetic series is necessary to understand how extensive the series is. This value is not only useful for conceptualizing the series length but is also plugged into the formula to find the sum. To find the number of terms:

• Take the difference between the last and first value of the variable used in the series (in our case, \( k \)), and add 1 to include the starting term itself.

Applying this to our example, with \( k = 11 \) and \( k = 7 \), the calculation becomes:

• \( n = (11 - 7) + 1 = 5 \).

This reveals that there are 5 terms in the sequence.

Common Difference

In an arithmetic series, the "common difference" is the constant gap between each pair of consecutive terms. It is important because it confirms the consistent growth or decline pattern in the series, verifying its arithmetic nature. To find the common difference:

• Subtract the first term from the second term.

For example, after substituting to find terms in our series:

• First term \( a_1 = -21 \) and second term \( a_2 = 42 - 9 \times 8 = -30 \).
• The common difference \( d \) is calculated as \( a_2 - a_1 = -30 - (-21) = -9 \).

This pattern repeats consistently with other terms.

Sum of Arithmetic Series

Calculating the sum of an arithmetic series packages all the concepts into a single expression that tells us the total of all the terms. The formula for the sum \( S_n \) is:

• \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( n \) is the number of terms and \( a_1 \) and \( a_n \) are the first and last terms, respectively.

Substituting the values from our example:

• With \( n = 5 \), \( a_1 = -21 \), and \( a_n = -57 \), the sum becomes \( S_5 = \frac{5}{2}(-21 + -57) \).
• This simplifies to \( \frac{5}{2}(-78) = -195 \).

Thus, the sum of the series is \(-195\), providing a neat conclusion to the sequence calculations.