Problem 21 - Algebra 2

Question

For each sum, find the number of terms, the first term, and the last term. Then evaluate the series. $$ \sum_{n=3}^{8}(7-n) $$

Step-by-Step Solution

Answer

The first term is 4, the last term is -1, the number of terms is 6, and the sum of the series is 9.

Step 1: Identify the first term

The first term of the series can be obtained by substituting n=3 into the equation, which gives us the result \(7-3 = 4\).

Step 2: Identify the last term

The last term of the series can be determined by substituting n=8 into the equation, which offers us the result of \(7-8 = -1\).

Step 3: Number of terms

With the series starting at n=3 and ending at n=8, the total number of terms is \(8 - 3 + 1 = 6\). Adding 1 is important because both the beginning and ending terms are inclusive.

Step 4: Calculate sum of the series

The sum of an arithmetic series can be computed using the formula: \[S = \frac{n}{2} * (a + l)\] where S is the sum of the series, n is the number of terms, a is the first term, and l is the last term. Substituting our values we get \[S = \frac{6}{2} * (4 + -1) = 3*3 = 9\].

Key Concepts

Sum of Arithmetic Sequence

The sum of an arithmetic sequence refers to adding together all the terms of the sequence. An arithmetic sequence is one in which each term after the first is obtained by adding a constant difference to the preceding term. To find the sum of an arithmetic sequence, you can use the formula: \[ S = \frac{n}{2} \times (a + l) \] Where:

• \( S \) is the sum of the sequence.
• \( n \) is the number of terms.
• \( a \) is the first term.
• \( l \) is the last term.

This formula essentially takes the average of the first and last term, and then multiplies it by the number of terms. This gives the total sum efficiently without having to manually add each term together. It is a powerful tool for finding sums quickly and accurately.

First and Last Terms

To evaluate an arithmetic sequence, it's crucial to identify the first and last terms explicitly, as they are key inputs for calculating the total sum. Finding the First Term: To find the first term, we substitute the starting value of \( n \) into the sequence equation. For example, substituting \( n = 3 \) in \( 7-n \) results in \( 4 \). This becomes our starting value of the series.Finding the Last Term: Similar to finding the first term, the last term is identified by substituting the ending value of \( n \) into the sequence equation. For instance, substituting \( n=8 \) gives us \( 7-8 = -1 \). These two values, first and last terms, are important for determining the sum of the sequence using the formula covered in the previous section.

Counting Terms in Series

In arithmetic sequences, understanding how to count the number of terms is critical. When a sequence is defined, such as with a range from \( n = 3 \) to \( n = 8 \), counting correctly ensures we interpret the sequence as intended.To find the number of terms, you use the formula: \[ \text{Number of terms} = (\text{last } n - \text{first } n) + 1 \] This method involves subtracting the first term index from the last term index and then adding one, because both boundaries, starting and ending terms, are included in the sequence.This approach ensures you accurately capture every element in the sequence from start to finish, providing a solid count to use when calculating the sum.

Evaluating Series

Evaluating an arithmetic series means calculating its sum by using various elements such as first and last terms, and the number of terms. Once these components are clearly identified, the sum is readily obtainable.Here is how you evaluate the series in a step-by-step manner:

• Identify each term within the range: Determine the first \( n \) and last \( n \).
• Calculate each term’s value: Substitute these into the sequence formula (i.e., \( 7-n \)).
• Count the number of terms: Use the formula \[ (\text{last } n - \text{first } n) + 1 \].
• Use the sum formula: Finally, apply \[ S = \frac{n}{2} \times (a + l) \], to find the sum.

By following these guidelines, you can efficiently and accurately determine the total of any arithmetic series, fully utilizing the powers of arithmetic sequences.