Problem 62 - Precalculus with Limits

Question

In Exercises 59 - 66, find the indicated \( n \)th partial sum of the arithmetic sequence. \( 0.5, 1.3, 2.1, 2.9, \cdots , n = 10 \)

Step-by-Step Solution

Answer

The 10th partial sum of the arithmetic sequence is 41.

Step 1: Identify the first term and the common difference

In the given arithmetic sequence, the first term (denoted as \(a\)) is 0.5 and the common difference (denoted as \(d\)) is obtained by subtracting second term by the first term i.e. \(d = 1.3 - 0.5 = 0.8\).

Step 2: Use the formula for the nth partial sum of an arithmetic sequence

The formula for the nth partial sum (\(S_n\)) of an arithmetic sequence is \(S_n = n/2 [2a + (n - 1)d]\). In this problem, we want to find the 10th partial sum, so n = 10. Plugging these values into the formula gives us \(S_{10} = 10/2 [ 2*0.5 + (10 - 1)*0.8 ]\).

Step 3: Calculate the sum

Simplify the expression to get the final answer. \(S_{10} = 5 [ 1 + 7.2 ] = 5 * 8.2 = 41\). Therefore, the 10th partial sum of the arithmetic sequence is 41.

Key Concepts

Common Difference in Arithmetic Sequence

Understanding the common difference in an arithmetic sequence is crucial for identifying the pattern within the series. The common difference, denoted as d, is the consistent interval between consecutive terms. To find the common difference, simply subtract the first term from the second term (or any term from the term that follows it).

For example, in our sequence: 0.5, 1.3, 2.1, 2.9,... the common difference is calculated by subtracting the first term from the second term:
\(d = 1.3 - 0.5 = 0.8\).

This value is consistent for every term in the sequence, meaning if we add 0.8 to the second term, we get the third term, and so on. This regular pattern is the defining characteristic of an arithmetic sequence and is the key to unlocking many problems related to arithmetic sequences, including finding specific terms and sums of terms.

Nth Term of Arithmetic Sequence

The nth term of an arithmetic sequence can be found by using a straightforward formula:
\(a_n = a + (n - 1)d\),
where a is the first term, d is the common difference, and n is the term number. This formula allows us to determine any term of the sequence without having to write out all the preceding terms.

For instance, in an arithmetic sequence with a first term of 0.5 and a common difference of 0.8, the 10th term would be found as follows:
\(a_{10} = 0.5 + (10 - 1) \times 0.8\).
By performing the arithmetic, \(a_{10} = 0.5 + 9 \times 0.8 = 0.5 + 7.2 = 7.7\).

The ability to calculate the nth term is especially useful when dealing with long sequences, as it provides a quick and efficient way to find specific terms without laborious counting.

Arithmetic Series

An arithmetic series is the summation of all terms in an arithmetic sequence. To find the sum of the first n terms of an arithmetic series, we use the formula:
\(S_n = \frac{n}{2} [2a + (n - 1)d]\),
where S_n is the nth partial sum, a is the first term, d is the common difference, and n is the number of terms.

Applying this formula to the exercise given, the sum of the first 10 terms of the sequence with a first term of 0.5 and a common difference of 0.8 is calculated as:
\(S_{10} = \frac{10}{2} [2 \times 0.5 + (10 - 1) \times 0.8]\).
Simplifying the terms inside the bracket and then multiplying by 5 yields the 10th partial sum:
\(S_{10} = 5 \times (1 + 7.2) = 5 \times 8.2 = 41\).

This approach to finding the sum is much more efficient than adding each term individually and is particularly advantageous for sequences with a large number of terms.