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 \(a_1\). In this sequence, the first term is \$0.0004$.
2. Determine the common ratio \(r\) by dividing the second term by the first term: \(r = \frac{-0.0004}{0.0004}\).
3. Write the general formula for the \(n\)th term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\).
4. Substitute the values of \(a_1\) and \(r\) into the formula to express \(a_n\) explicitly for this sequence.
5. To find the seventh term \(a_7\), substitute \(n = 7\) into the formula: \(a_7 = a_1 \times r^{6}\).

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, and it determines the pattern of growth or decay in the sequence.

General Term Formula of a Geometric Sequence

The general term (nth term) of a geometric sequence is given by aₙ = a₁ * r^(n-1), where a₁ is the first term, r is the common ratio, and n is the term number. 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 specific term by substituting the term number n into the formula. For example, to find the seventh term a₇, plug in n = 7 and calculate using the known first term and common ratio.