In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a₇, the seventh term of the sequence. 0.0004, - 0.0004, 0.04, - 0.04, ...

Verified step by step guidance

1. Identify the first term of the geometric sequence, denoted as $a=0.0004$.
2. Determine the common ratio $r$ by dividing the second term by the first term: $r\; =\; \backslash frac\{-0.0004\}\{0.0004\}\; =\; -1$. Verify this ratio with subsequent terms to confirm consistency.
3. Write the general formula for the nth term of a geometric sequence: $a\_n\; =\; a\; \backslash cdot\; r^\{n-1\}$.
4. Substitute the values of $a$ and $r$ into the formula to get $a\_n\; =\; 0.0004\; \backslash cdot\; (-1)^\{n-1\}$.
5. To find the seventh term $a\_7$, substitute $n\; =\; 7$ into the formula: $a\_7\; =\; 0.0004\; \backslash cdot\; (-1)^\{7-1\}$. Simplify the exponent expression to proceed.

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. This ratio can be positive or negative, affecting the sign and magnitude of the terms.

General Term Formula of a Geometric Sequence

The general term (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 allows you to find any term in the sequence without listing all previous terms.

Finding Specific Terms Using the General Formula

Once the general term formula is established, you can find any term by substituting the term number n into the formula. For example, to find the seventh term, substitute n = 7 into a_n to calculate a_7 directly.