Find a₂ and a₃ for each geometric sequence. 8, a₂, a₃, 27

Verified step by step guidance

1. Identify the first term of the geometric sequence as \(a_1 = 8\) and the fourth term as \(a_4 = 27\).
2. Recall the formula for the \(n\)th term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\), where \(r\) is the common ratio.
3. Use the formula for the fourth term to set up the equation: \(a_4 = a_1 \times r^{3}\), which becomes \$27 = 8 \times r^{3}$.
4. Solve for the common ratio \(r\) by dividing both sides by 8 and then taking the cube root: \(r^{3} = \frac{27}{8}\), so \(r = \sqrt[3]{\frac{27}{8}}\).
5. Find \(a_2\) and \(a_3\) using the formula \(a_n = a_1 \times r^{n-1}\): calculate \(a_2 = 8 \times r\) and \(a_3 = 8 \times r^{2}\).

Key Concepts

Geometric Sequence

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. For example, in the sequence 2, 6, 18, 54, the common ratio is 3.

Common Ratio

The common ratio in a geometric sequence is the fixed factor between consecutive terms. It can be found by dividing any term by its preceding term. Knowing the common ratio allows you to find missing terms in the sequence.

Finding Missing Terms in a Sequence

To find missing terms in a geometric sequence, use the formula a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio. By setting up equations with known terms, you can solve for unknown terms like a_2 and a_3.