Problem 54 - Precalculus with Limits

Question

In Exercises 51 - 58, find the sum of the finite arithmetic sequence. \( -5 + \left(-3\right) + \left(-1\right) + 1 + 3 + 5 \)

Step-by-Step Solution

Answer

The sum of the arithmetic sequence is 0.

Step 1: Identify the first and the last term

Identify the first and the last term in the sequence. The first term \( a_1 \) is -5 and the last term \( a_n \) is 5.

Step 2: Determine the number of terms

Count the number of terms in the sequence. There are six terms so \( n = 6 \)

Step 3: Apply the formula

Use the formula for the sum of an arithmetic series and substitute the values. The sum is, \( S = \frac{6}{2} \left( -5 + 5 \right) = 0 \)

Step 4: Simplify the expression

Simplify the expression to get the sum of the sequence. The sum of the sequence is Zero.

Key Concepts

Sum of Arithmetic Sequences

In an arithmetic sequence, each term increases by a constant amount, known as the common difference. The sum of a finite arithmetic sequence or series involves adding up all the terms in the sequence. Calculating this sum can be more efficient using the formula for an arithmetic series, rather than adding each term individually. This is especially useful when dealing with a large number of terms.

Consider the sequence from the original exercise:

• First term: \( a_1 = -5 \)
• Last term: \( a_n = 5 \)
• Number of terms: \( n = 6 \)

The sum of this sequence is determined using a specific formula, which will be explained further in the next section.

Arithmetic Series Formula

The arithmetic series formula is a powerful tool used to find the sum of any arithmetic sequence quickly. The general formula for the sum of the first \( n \) terms of an arithmetic sequence is:\[ S_n = \frac{n}{2} (a_1 + a_n)\]Where:

• \( S_n \) is the sum of the sequence.
• \( n \) is the number of terms in the sequence.
• \( a_1 \) is the first term of the sequence.
• \( a_n \) is the last term of the sequence.

To find the sum, you simply need to find the first term, last term, and the total number of terms. For the sequence given, using the formula, we have:\[ S = \frac{6}{2} (-5 + 5) = 0\]This shows the sum of the sequence as zero, demonstrating how effective the formula is even with sequences containing negative numbers.

Number of Terms in a Sequence

Determining the number of terms in an arithmetic sequence is essential, especially when using the arithmetic series formula. Knowing the total number of terms allows you to apply the formula correctly.

In our example sequence:

• The terms listed are: -5, -3, -1, 1, 3, 5.
• Counting these gives us a total of 6 terms.

The number of terms can also be found using the formula:\[n = \frac{a_n - a_1}{d} + 1\]where \( d \) is the common difference between terms. Here the common difference \( d \) is 2, confirming again that \( n = 6 \). This makes using the arithmetic series formula possible, ensuring both accuracy and efficiency.