Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{3}=5, a_{4}=8$$

Step-by-Step Solution

Answer

The sum of the first 10 terms is 125.

Step 1: Understand Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference and is denoted by \(d\).

Step 2: Identify Known Terms

The third term of the sequence is \( a_3 = 5 \) and the fourth term is \( a_4 = 8 \).

Step 3: Determine Common Difference

Calculate the common difference \(d\) by subtracting the third term from the fourth term: \( d = a_4 - a_3 = 8 - 5 = 3 \). Thus, the common difference \(d\) is 3.

Step 4: Find First Term (a1)

We use the formula for the nth term of an arithmetic sequence: \( a_n = a_1 + (n-1)d \).For the third term:\( a_3 = a_1 + 2d = 5 \).Substitute \(d = 3\):\( a_1 + 2(3) = 5 \).Simplify:\( a_1 + 6 = 5 \) \( a_1 = 5 - 6 = -1 \). So, the first term \(a_1\) is -1.

Step 5: Sum Formula for Arithmetic Sequence

The sum of the first \(n\) terms of an arithmetic sequence is given by: \( S_n = \frac{n}{2} (a_1 + a_n) \) where \(a_n\) is the nth term. For the first 10 terms, substitute \( n = 10 \).

Step 6: Find the 10th Term (a10)

To find \(a_{10}\), use the formula for the nth term again: \( a_{10} = a_1 + 9d = -1 + 9(3) = 26 \). Hence, \(a_{10} = 26\).

Step 7: Calculate the Sum of First 10 Terms

Using the sum formula: \( S_{10} = \frac{10}{2} (a_1 + a_{10}) = 5(-1 + 26) \).Calculate: \( S_{10} = 5(25) = 125 \). Therefore, the sum of the first 10 terms is 125.

Key Concepts

Common Difference

In an arithmetic sequence, one of the core aspects is the common difference. This is the constant amount you add to each term to get to the next term in the sequence. Mathematically, it is represented by the symbol \(d\).

For instance, if we know two consecutive terms, we can find the common difference by subtracting the first of these terms from the next. For our example, with \(a_3 = 5\) and \(a_4 = 8\), the common difference \(d\) is calculated as \(d = a_4 - a_3 = 8 - 5 = 3\).

Once the common difference is found, it helps in predicting any other terms in the sequence and plays a crucial role in using any arithmetic sequence formula. Remember that the common difference can be positive, negative, or zero, depending on the sequence.

Sum of Terms

The sum of terms in an arithmetic sequence is a frequent computation, particularly when dealing with a large number of terms. The formula used to compute the sum of the first \(n\) terms is:

• \(S_n = \frac{n}{2} (a_1 + a_n)\)

Here, \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term of the sequence, and \(a_n\) is the nth term.

In our example, to find the sum of the first 10 terms, we need to calculate \(a_1\), which is -1, and \(a_{10}\), which is 26, using the formula for the nth term. Substituting these into the sum formula gives:

• \(S_{10} = \frac{10}{2} (a_1 + a_{10}) = 5(-1 + 26)\)
• \(S_{10} = 5 \times 25 = 125\)

This indicates that the sum of the first 10 terms is 125.

Nth Term Formula

The nth term formula is a powerful tool in arithmetic sequences. It allows you to find any term in the sequence if you know the first term \(a_1\) and the common difference \(d\). The formula is:

• \(a_n = a_1 + (n-1)d\)

This formula computes the nth term directly without the need to list all the preceding terms.

For example, to find the 10th term \(a_{10}\) of the sequence, with \(a_1 = -1\) and \(d = 3\), we compute:

• \(a_{10} = a_1 + 9d = -1 + 9\times 3 = -1 + 27 = 26\)

Thus, using the nth term formula simplifies calculations, making it possible to quickly find the desired term in the sequence.