Problem 60 - Precalculus with Limits

Question

In Exercises 59 - 66, find the indicated \( n \)th partial sum of the arithmetic sequence. \( -6, -2, 2, 6, \cdots , n = 50 \)

Step-by-Step Solution

Answer

The 50th partial sum of the given arithmetic sequence is 9300

Step 1: Determine the common difference

The common difference of an arithmetic sequence is found by subtracting any two consecutive terms. Let's take the second term (-2) and subtract the first term (-6) from it. The result is 4, which is the common difference.

Step 2: Compute the nth term

The nth term of an arithmetic sequence is given by \( a + (n-1) * d \), where 'a' is the first term, 'n' is the term number and 'd' is the common difference. For this problem, 'a' is -6, 'n' is 50, and 'd' is 4. Substituting these values in we get, \( a_50 = -6 + (50-1)*4 = 192 \)

Step 3: Compute the nth partial sum

The nth partial sum of an arithmetic sequence is given by \( n/2 * (a + a_n) \), where 'a' is the first term, 'a_n' is the nth term, and 'n' is the term number. Here, 'a' is -6, 'a_n' is 192 and 'n' is 50. Substituting these values in we get, \( S_{50} = 50/2 * (-6 + 192) = 9300 \)

Key Concepts

Common Difference

In an arithmetic sequence, the difference between each pair of consecutive terms is always the same. This constant difference is known as the "common difference." To find this difference, one can simply subtract any term from the term that follows it.
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For example, consider an arithmetic sequence: \(-6, -2, 2, 6, \ldots\). By subtracting the first term \(-6\) from the second term \(-2\), we can see that:

• -2 - (-6) = 4

This calculation shows that the common difference in this sequence is 4. Understanding the common difference helps predict all subsequent terms in the sequence, as it provides a simple rule to follow. Each term is simply the result of adding the common difference to the term before it.
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Knowing the common difference is essential for finding other elements within an arithmetic sequence, such as the nth term or the partial sum.

Partial Sum

The partial sum of an arithmetic sequence refers to the sum of the first "n" terms of that sequence. It's a way to compute how much those initial terms add up to when added together. Calculating this can be especially helpful in many mathematical and practical tasks, such as finding out how much of a particular resource is needed for several stages of a process.
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The formula for finding the partial sum \(S_n\) of an arithmetic sequence is:

• \( S_n = \frac{n}{2} \times (a + a_n) \)

Here, "\(n\)" represents the number of terms to add up, "\(a\)" is the first term in the sequence, and "\(a_n\)" is the nth term.
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For instance, in our given sequence \(-6, -2, 2, 6, \ldots\), to find the partial sum for the first 50 terms, we substitute the values into the formula:

• \( S_{50} = \frac{50}{2} \times (-6 + 192) = 9300 \)

Thus, the sum of the first 50 terms of this arithmetic sequence is 9300.

nth Term of an Arithmetic Sequence

Each term in an arithmetic sequence can be expressed as a function of its position or its "n" value. This is what we refer to as the "nth term" of an arithmetic sequence. It allows us to find any term in the sequence without writing out the entire series of numbers leading up to it.
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The formula to find the nth term, \(a_n\), is given by:

• \( a_n = a + (n-1) \times d \)

Where "\(a\)" is the first term, "\(d\)" is the common difference, and "\(n\)" is the position of the term in the sequence.
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For example, to find the 50th term of the sequence \(-6, -2, 2, 6, \ldots\), we apply the formula:

• \( a_{50} = -6 + (50-1) \times 4 = 192 \)

This tells us that the 50th term is 192.
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This formula makes it easy to determine any term in the sequence quickly and efficiently, using just the common difference and the position of the term in the sequence.