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. a_n = a_n-1 +3, a₁ = 4

Verified step by step guidance

1. Identify the first term of the arithmetic sequence, which is given as \(a_1 = 4\).
2. Recognize that the sequence increases by a common difference of 3 each time, as indicated by the recursive formula \(a_n = a_{n-1} + 3\).
3. Write the explicit formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n - 1)d\), where \(d\) is the common difference.
4. Substitute the known values into the formula: \(a_n = 4 + (n - 1) \times 3\).
5. To find the 20th term, substitute \(n = 20\) into the explicit formula: \(a_{20} = 4 + (20 - 1) \times 3\).

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 difference is called the common difference. For example, in the sequence 4, 7, 10, 13, ..., the common difference is 3.

General Term Formula of an Arithmetic Sequence

The general term (nth term) of an arithmetic sequence can be expressed as an = a₁ + (n - 1)d, where a₁ 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.

Recursive vs Explicit Formulas

A recursive formula defines each term based on the previous term(s), while an explicit formula gives a direct expression for the nth term. Converting a recursive formula like an = an-1 + 3 into an explicit formula simplifies finding specific terms, such as the 20th term.