In Exercises 63–64, find a2 and a3 for each geometric sequence. 2, a₂, a₃, - 54

Verified step by step guidance

1. Identify the first term of the geometric sequence, which is given as $a{1}_{}=2$.
2. Recall that in a geometric sequence, each term is found by multiplying the previous term by the common ratio $r$. So, $a\_n\; =\; a\_1\; imes\; r^\{n-1\}$.
3. Use the information about the fourth term to find the common ratio. The fourth term is given as $a\_4\; =\; -54$. Using the formula, $a\_4\; =\; a\_1\; imes\; r^\{3\}$, substitute the known values: $-54\; =\; 2\; imes\; r^\{3\}$.
4. Solve the equation $-54\; =\; 2\; imes\; r^\{3\}$ for $r^\{3\}$ by dividing both sides by 2, giving $r^\{3\}\; =\; -27$. Then find $r$ by taking the cube root of both sides.
5. Once you have the value of $r$, find $a\_2$ and $a\_3$ using the formulas $a\_2\; =\; a\_1\; imes\; r$ and $a\_3\; =\; a\_1\; imes\; r^\{2\}$.

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 when some terms are given.

Term Formula for Geometric Sequences

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