Find a₂ and a₃ for each geometric sequence. 2, a₂, a₃, - 54

Verified step by step guidance

1. Identify the terms of the geometric sequence given: the first term \(a_1 = 2\), the second term \(a_2\), the third term \(a_3\), and the fourth term \(a_4 = -54\).
2. Recall that in a geometric sequence, each term is found by multiplying the previous term by a common ratio \(r\). So, \(a_2 = a_1 \times r\), \(a_3 = a_2 \times r = a_1 \times r^2\), and \(a_4 = a_1 \times r^3\).
3. Use the known value of the fourth term to set up an equation: \(a_4 = a_1 \times r^3 = -54\). Substitute \(a_1 = 2\) to get \(2 \times r^3 = -54\).
4. Solve the equation \(2 \times r^3 = -54\) for \(r^3\) by dividing both sides by 2, then find \(r\) by taking the cube root of both sides.
5. Once you have the value of \(r\), calculate \(a_2\) using \(a_2 = 2 \times r\) and \(a_3\) using \(a_3 = 2 \times r^2\).

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. Understanding this definition helps identify the relationship between consecutive terms.

Common Ratio Calculation

The common ratio (r) is found by dividing any term by its preceding term. Knowing how to calculate r allows you to find unknown terms in the sequence by multiplying the previous term by r.

Term Formula for Geometric Sequences

The nth term of a geometric sequence is given by a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio. This formula is essential for finding specific terms like a2 and a3 when some terms are known.