Problem 43 - Precalculus Mathematics for Calculus

Question

Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a=1, d=2, n=10$$

Step-by-Step Solution

Answer

The partial sum \(S_{10}\) is 100.

Step 1: Understand What is Given

You are given the first term of an arithmetic sequence \(a = 1\), the common difference \(d = 2\), and the number of terms \(n = 10\). Our goal is to find the 10th partial sum \(S_{10}\).

Step 2: Recall the Formula for Partial Sum

The formula for the partial sum \(S_n\) of an arithmetic sequence is given by \[S_n = \frac{n}{2} (2a + (n-1)d)\] where \(n\) is the number of terms, \(a\) is the first term, and \(d\) is the common difference.

Step 3: Substitute Known Values into the Formula

Substitute the known values into the formula: \[S_{10} = \frac{10}{2} (2 \times 1 + (10-1) \times 2)\]

Step 4: Simplify the Expression

First, calculate inside the parentheses:\(2 \times 1 = 2\) and \((10-1) \times 2 = 18\). Adding these results gives:\[2 + 18 = 20\]

Step 5: Calculate the Partial Sum

Substitute back into the partial sum equation:\[S_{10} = \frac{10}{2} \times 20\]This simplifies to:\[S_{10} = 5 \times 20 = 100\]

Step 6: Verify the Calculation

To verify, consider calculating the sum by listing terms of the sequence: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.Adding these terms: \(1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100\).This confirms the calculated partial sum is correct.

Key Concepts

Partial Sum

The concept of a partial sum in an arithmetic sequence is crucial to understanding how to find the sum of a series of numbers. Imagine you have a set of numbers arranged in a specific order, and you want to know their collective total up to a certain point. This total is known as a partial sum. Here, we look at the first several terms and add them together. In the context of the exercise, the partial sum we are interested in is for the first 10 terms. By calculating this sum, we find out the total value of those terms added together. In arithmetic sequences, there’s a handy formula to speed up this process, which allows us to find the sum without having to manually add every single term. In the example given, the partial sum was calculated to be 100, showing just how powerful and efficient using a formula can be.

Common Difference

The common difference in an arithmetic sequence is what makes these sequences predictable and easy to work with. This term refers to the consistent interval between consecutive numbers in the sequence. To identify the common difference, simply subtract any term in the sequence from the one that follows it. In our example, starting with the first term of 1 and a next term of 3, the common difference is 2. This means that each number in the sequence increases by 2 as you move to the next term. Understanding the common difference is vital because it allows you to predict future terms of the sequence and forms a part of the formula needed to determine the partial sum.

First Term

The first term of an arithmetic sequence is where everything begins. It lays the foundation for the entire sequence. Knowing this term is essential because it is used to calculate subsequent terms through the addition of the common difference. In the given exercise, the first term is 1. This is the starting point of our sequence, and each following number builds upon it by adding the common difference. When looking for partial sums or trying to find a specific term in the sequence, you always refer back to the first term. It’s a key ingredient in the formula to find partial sums as well, appearing clearly as part of the calculation method we use.

Formula for Partial Sum

The formula for calculating the partial sum of an arithmetic sequence is extremely useful, especially when dealing with longer sequences. It allows us to find the sum of a number of terms without summing each term individually.For an arithmetic sequence, the formula is: \[S_n = \frac{n}{2} (2a + (n-1)d)\]where

• \(S_n\) is the partial sum of the first \(n\) terms,
• \(a\) is the first term,
• \(d\) is the common difference,
• \(n\) is the number of terms being summed.

Using this formula, we can efficiently calculate the partial sum. In our specific example, substituting the known values into the formula allowed us to find the partial sum for 10 terms quickly, arriving at a result of 100. This formula is not only powerful for saving time, but also for ensuring accuracy in calculations.