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. Calculate the common difference \(d\) by subtracting the first term from the second term: \(d = -3 - (-7) = -3 + 7\).
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 \(a_n\).
5. Use the formula to find the 20th term by substituting \(n = 20\) into the expression for \(a_n\).

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. For example, in the sequence -7, -3, 1, 5, the common difference is 4. Understanding this pattern is essential to formulating the general term.

General Term Formula of an Arithmetic Sequence

The general term (nth term) of an arithmetic sequence is given by an = a1 + (n - 1)d, where a1 is the first term and d is the common difference. This formula allows direct calculation of any term without listing all previous terms.

Evaluating the nth Term

Once the general term formula is established, substituting n with the desired term number (like 20 for a20) gives the value of that term. This step involves simple arithmetic and is crucial for finding specific terms efficiently.