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 -10, a₁ = 30

Verified step by step guidance

1. Identify the first term of the arithmetic sequence, which is given as \(a_1 = 30\).
2. Recognize that the recursive formula \(a_n = a_{n-1} - 10\) indicates a common difference \(d = -10\) because each term decreases by 10 from the previous term.
3. Write the explicit formula for the nth term of an arithmetic sequence using the formula: \[a_n = a_1 + (n - 1) \times d\]
4. Substitute the known values \(a_1 = 30\) and \(d = -10\) into the formula to get: \[a_n = 30 + (n - 1) \times (-10)\]
5. To find the 20th term, substitute \(n = 20\) into the explicit formula: \[a_{20} = 30 + (20 - 1) \times (-10)\] Then simplify to find the value of \(a_{20}\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is found by adding or subtracting a constant difference. This difference remains the same throughout the sequence, making it predictable and linear.

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 and d is the common difference. This formula allows direct calculation of any term without recursion.

Using the General Term to Find Specific Terms

Once the general term formula is established, it can be used to find any term in the sequence by substituting the term number n. For example, to find the 20th term, substitute n = 20 into the formula.