In Exercises 23–34, 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 to find the 20th term of the sequence. 1, 5, 9, 13,...

Verified step by step guidance

1. insert step 1: Identify the first term (a_1) of the sequence. In this case, a_1 = 1.
2. insert step 2: Determine the common difference (d) by subtracting the first term from the second term. Here, d = 5 - 1 = 4.
3. insert step 3: Write the formula for the nth term of an arithmetic sequence: a_n = a_1 + (n - 1) * d.
4. insert step 4: Substitute the known values into the formula: a_n = 1 + (n - 1) * 4.
5. insert step 5: To find the 20th term, substitute n = 20 into the formula: a_20 = 1 + (20 - 1) * 4.

Key Concepts

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. For example, in the sequence 1, 5, 9, 13, the common difference is 4, as each term increases by 4 from the previous term.

General Term Formula

The general term formula for an arithmetic sequence allows us to find any term in the sequence without having to list all previous terms. It is typically expressed as a_n = a_1 + (n - 1)d, where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference.

Finding Specific Terms

To find a specific term in an arithmetic sequence using the general term formula, substitute the desired term number into the formula. For instance, to find the 20th term of the sequence 1, 5, 9, 13, you would use the formula a_n = 1 + (20 - 1) * 4, which simplifies to calculate the value of the 20th term.