In Exercises 63–64, 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, which is $a{1}_{}=8$.
2. Recognize that in a geometric sequence, each term is found by multiplying the previous term by the common ratio $r$. So, $a{n}_{}=a{1}_{}\cdot r{\mathrm{n-1}}^{}$.
3. Use the given fourth term $a{4}_{}=\; 27$ to set up the equation $27\; =\; 8\; imes\; r^\{3\}$ because the fourth term is the first term times $r$ cubed.
4. Solve for the common ratio $r$ by dividing both sides by 8 and then taking the cube root: .
5. Once you find $r$, calculate $a\_2\; =\; 8\; imes\; r$ and $a\_3\; =\; 8\; imes\; r^\{2\}$ to find the second and third terms of the sequence.

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. In this problem, knowing the first and fourth terms allows you to calculate r by using the formula a_n = a_1 * r^(n-1).

Term Formula for Geometric Sequences

The nth term of a geometric sequence is given by a_n = a_1 * r^(n-1). Using this formula, you can find any term in the sequence once the first term and common ratio are known.