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 a₂₀, the 20th term of the sequence. 7,3,-1,-5,...

Verified step by step guidance

1. Identify the first term of the arithmetic sequence, which is given as \(a_1 = 7\).
2. Determine the common difference \(d\) by subtracting the first term from the second term: \(d = 3 - 7\).
3. Write the general formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1) \times d\).
4. Substitute the values of \(a_1\) and \(d\) into the formula to get the explicit formula for the nth term.
5. Use the formula to find the 20th term by substituting \(n = 20\) into the formula.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference to the previous term. This constant is called the common difference. For example, in the sequence 7, 3, -1, -5, the common difference is -4.

General Term Formula of an Arithmetic Sequence

The general term (nth term) of an arithmetic sequence can be expressed as a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is the term number. This formula allows direct calculation of any term without recursion.

Evaluating the nth Term

Once the general term formula is established, substitute the desired term number (n) into the formula to find that term's value. For example, to find the 20th term, plug n = 20 into the formula and simplify to get the result.