Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the indicated \(n\) th partial sum of the arithmetic sequence. $$8,20,32,44, \ldots ; n=10$$

Step-by-Step Solution

Answer

The 10th partial sum of the given arithmetic sequence is 620.

Step 1: Identify the first term, the number of terms, and the common difference

The first term \(a\) in the arithmetic sequence is 8. The number of terms \(n\) to consider for the partial sum is given as 10. The common difference \(d\) can be determined by subtracting any term from the term that follows it, which is 12 in this case.

Step 2: Calculate the last term

The last term \(l\) of the first 10 terms in the sequence can be found using the formula \(l = a + (n-1)d\). Substituting the values, we get: \(l = 8 + (10 - 1)12 = 116\)

Step 3: Calculate the 10th partial sum

Use the formula for the \(n\)th partial sum of an arithmetic sequence, \(S_n = n/2 * (a + l)\). Substitute in the values we know: \(S_{10} = 10/2 * (8 + 116) = 5 * 124 = 620.

Key Concepts

Partial Sum

In an arithmetic sequence, the partial sum refers to the sum of a certain number of terms from the start of the sequence up to a specified point. It is essentially the sum of the first few numbers that define the sequence.
For arithmetic sequences, there is a simple formula to find the partial sum: \[ S_n = \frac{n}{2}(a + l) \]where:

• \( S_n \) is the partial sum
• \( n \) is the number of terms
• \( a \) is the first term
• \( l \) is the last term

Calculating the partial sum allows you to find out how much all those numbers add up to together without adding each one individually, which saves time when dealing with long sequences. In the problem, the partial sum of the first 10 terms of the sequence is 620.

Common Difference

The common difference in an arithmetic sequence is the consistent difference between consecutive terms. It is what makes the sequence "arithmetic," meaning that each term increases or decreases by the same amount from the previous term.
You can find the common difference by subtracting any term from the term that immediately follows it. For example, with the sequence 8, 20, 32, 44, ..., the common difference \( d \) is determined by: \[ d = 20 - 8 = 12 \]The common difference of 12 tells us that each term is 12 more than the term before it.
Understanding this concept is crucial because it allows us to predict future terms in the sequence and is also necessary for calculating the partial sum.

First Term

The first term in an arithmetic sequence is the starting point of the sequence—the very first number from which all other terms are derived. It is crucial because it anchors the entire sequence and is a key component of the formula used to find other aspects of the sequence, such as its partial sum.
In our example with the sequence 8, 20, 32, 44, ..., the first term \( a \) is 8. This initial value is important because it combines with the common difference to define the shape and direction of the entire sequence.
Often, knowing the first term allows you to find any other specific term or even the entire sequence, provided you also know the common difference.