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. an = an-1 +3, a₁ = 4

Verified step by step guidance

1. Identify the first term $a_1$ of the sequence, which is given as 4.
2. Recognize that the sequence is arithmetic with a common difference $d$ of 3.
3. Use the formula for the nth term of an arithmetic sequence: $a_n=a_1+(n-1)\cdot d$.
4. Substitute the known values into the formula: $a_n=4+(n-1)\cdot 3$.
5. To find the 20th term, substitute $n=20$ into the formula: $a_{20}=4+(20-1)\cdot 3$.

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. For example, in the sequence 4, 7, 10, 13, the common difference is 3. Understanding this concept is crucial for deriving the general term of the sequence.

General Term Formula

The general term formula for 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 us to calculate any term in the sequence without needing to know the previous terms. For the given sequence, substituting the values will yield the formula for the nth term.

Finding Specific Terms

To find a specific term in an arithmetic sequence using the general term formula, substitute the desired term number (n) into the formula. For instance, to find the 20th term, you would replace n with 20 in the general term formula derived earlier. This process illustrates how to efficiently compute terms in a sequence without recursion.