In Exercises 1–14, write the first six terms of each arithmetic sequence. an = an-1 +6, a₁ = −9

Verified step by step guidance

1. Identify the first term of the sequence, which is given as \( a_1 = -9 \).
2. Recognize that the sequence is arithmetic, meaning each term is obtained by adding a constant difference to the previous term. Here, the common difference \( d \) is 6.
3. Use the recursive formula \( a_n = a_{n-1} + 6 \) to find the second term: \( a_2 = a_1 + 6 \).
4. Continue using the recursive formula to find the third term: \( a_3 = a_2 + 6 \).
5. Repeat the process to find the fourth, fifth, and sixth terms: \( a_4 = a_3 + 6 \), \( a_5 = a_4 + 6 \), and \( a_6 = a_5 + 6 \).

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. In the given sequence, the common difference is 6, meaning each term is obtained by adding 6 to the previous term.

Recursive Formula

A recursive formula defines each term of a sequence based on the preceding term(s). In this case, the formula an = an-1 + 6 indicates that each term (an) is derived from the previous term (an-1) by adding 6. This approach is essential for generating terms in sequences where a direct formula is not provided.

Initial Term

The initial term of a sequence is the first term from which all subsequent terms are generated. In this problem, the initial term is given as a₁ = -9. This starting point is crucial for calculating the first six terms of the arithmetic sequence, as it sets the foundation for all following terms.