Problem 35 - College Algebra

Question

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. Find the sum of the first 20 terms of the arithmetic sequence: \(4,10,16,22, \dots\)

Step-by-Step Solution

Answer

The 20th term of the arithmetic sequence is 118 and the sum of the first 20 terms is 1220.

Step 1: Find the common difference

The common difference, d, is the difference between any two successive terms. Here, d = 10 - 4 = 6.

Step 2: Find the nth term

Using the formula for the nth term of an arithmetic sequence, \(a_{n}=a_{1}+d(n-1)\), where \(a_{1}\) is the first term (in this case, 4), d is the common difference (6) and n is the term number, we get: \(a_{20}=4+6(20-1)=4+6*19=118\). So the 20th term of this arithmetic sequence is 118.

Step 3: Find the sum of the first n terms

Now using the formula for the sum of the first n terms of an arithmetic sequence, \(S_{n}=\frac{n}{2}(a_{1}+a_{n})\), where \(S_{n}\) is the sum of the first n terms, \(a_{1}\) is the first term (4) and \(a_{n}\) is the nth term (118), we get: \(S_{20}=\frac{20}{2}(4+118)=10*122=1220\). So the sum of the first 20 terms of this arithmetic sequence is 1220.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a constant value that is added to each term to get the next term. Imagine you have a series of numbers and each one is obtained by adding a certain number to the previous one. This number, which remains the same throughout the sequence, is called the common difference and is usually denoted by the letter \(d\). For example, in the sequence 4, 10, 16, 22, ..., the common difference is 6. We calculate it by subtracting the first term from the second term:

• Common Difference \(d = 10 - 4 = 6 \)
• It can be checked with any two successive terms: \(16 - 10 = 6\)

Understanding this helps in determining the pace at which the numbers in the sequence increase or decrease.

nth Term Formula

The nth term formula is a way to find any term in an arithmetic sequence without having to list all the terms. This formula is useful because it allows you to directly calculate the value of the nth term using the first term, the common difference, and the term number \(n\). The formula for finding the nth term \(a_n\) is given by:

• \(a_n = a_1 + d(n-1)\)

Where:

• \(a_1\) is the first term of the sequence
• \(d\) is the common difference
• \(n\) is the term number you're looking for

For example, to find the 20th term in our sequence, we use the formula:

• \(a_{20} = 4 + 6(20-1) = 4 + 114 = 118\)

This means the 20th term of the sequence is 118. This formula gives a simple method to pinpoint any term in the sequence.

Sum of Arithmetic Series

The sum of an arithmetic series refers to the sum of the terms in the sequence up to a certain point. Calculating this sum becomes easy with a clever formula instead of adding each term manually. The formula for 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, and \(a_n\) is the nth term. This formula can be broken down as taking the average of the first and last term and multiplying it by the number of terms. For the sequence 4, 10, 16, 22, ..., to find the sum of the first 20 terms, you would compute:

• \(S_{20} = \frac{20}{2} (4 + 118) = 10 \times 122 = 1220\)

So, the sum of the first 20 terms is 1220. This formula is efficient and saves time.

General Term of Sequence

The general term of a sequence is a formula that helps us define any individual term in a sequence with respect to its position number. In arithmetic sequences, this is achieved using the nth term formula, as it provides a clear rule to follow for finding terms. By using the general term formula \(a_n = a_1 + d(n-1)\), we can determine any specific term we want. This formula does not require finding every preceding term, making it extremely convenient for larger sequences or when specific terms far into the sequence need to be found. Being able to generate or derive such formulas for any sequence helps in identifying patterns and in calculating the sequence's behavior quickly. The general term hence encapsulates the entire sequence pattern in a single mathematical expression.