Write the first five terms of each geometric sequence. a_n = - 5a_(n-1), a₁ = - 6

Verified step by step guidance

1. Identify the first term of the geometric sequence, which is given as \(a_1 = -6\).
2. Recognize the recursive formula for the sequence: \(a_n = -5a_{n-1}\), meaning each term is obtained by multiplying the previous term by \(-5\).
3. Calculate the second term by multiplying the first term by \(-5\): \(a_2 = -5 \times a_1\).
4. Find the third term by multiplying the second term by \(-5\): \(a_3 = -5 \times a_2\).
5. Continue this process to find the fourth and fifth terms: \(a_4 = -5 \times a_3\) and \(a_5 = -5 \times a_4\).

Key Concepts

Geometric Sequence Definition

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This ratio determines how the sequence progresses, either growing or shrinking exponentially.

Recursive Formula for Sequences

A recursive formula defines each term of a sequence based on the previous term(s). In this problem, the term a_n depends on a_(n-1) multiplied by -5, which means each term is generated by applying this rule starting from the initial term.

Calculating Terms of a Sequence

To find terms in a sequence using a recursive formula, start with the given first term and repeatedly apply the formula to find subsequent terms. For example, multiply the first term by the common ratio to get the second term, then continue this process to find the next terms.