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. a1 = 9, d=2

Verified step by step guidance

1. Identify the formula for the nth term of an arithmetic sequence: $a_n = a_1 + (n-1) \cdot d$.
2. Substitute the given values into the formula: $a_1 = 9$ and $d = 2$.
3. The formula becomes: $a_n = 9 + (n-1) \cdot 2$.
4. Simplify the expression: $a_n = 9 + 2n - 2$.
5. Combine like terms to get the general formula: $a_n = 2n + 7$.

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 (d). For example, in the sequence 9, 11, 13, 15, the common difference is 2. Understanding this concept is crucial for identifying the pattern in the sequence and deriving the general term.

General Term Formula

The general term (nth term) of an arithmetic sequence can be expressed using the formula a_n = a_1 + (n - 1)d, where a_n is the nth term, a_1 is the first term, d is the common difference, and n is the term number. This formula allows us to calculate any term in the sequence without needing to list all previous terms, making it essential for solving problems related to arithmetic sequences.

Finding Specific Terms

To find a specific term in an arithmetic sequence, such as the 20th term, you substitute the term number (n) into the general term formula. For instance, using the previously mentioned formula with a_1 = 9 and d = 2, you would calculate a_20 = 9 + (20 - 1) * 2. This process illustrates how to apply the general term formula to derive specific values from the sequence.