Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a₁ and common ratio, r. Find a₄₀ when a₁ = 1000, r = - 1/2

Verified step by step guidance

1. Recall the formula for the nth term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
2. Identify the given values: the first term \(a_1 = 1000\), the common ratio \(r = -\frac{1}{2}\), and the term to find is \(a_{40}\), so \(n = 40\).
3. Substitute the known values into the formula: \(a_{40} = 1000 \times \left(-\frac{1}{2}\right)^{40-1}\).
4. Simplify the exponent expression: \(a_{40} = 1000 \times \left(-\frac{1}{2}\right)^{39}\).
5. Evaluate the power \(\left(-\frac{1}{2}\right)^{39}\) carefully, noting that since 39 is odd, the result will be negative, and then multiply by 1000 to find \(a_{40}\).

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, negative, or a fraction, affecting the sequence's behavior. Understanding this helps identify how terms progress.

General Term Formula of a Geometric Sequence

The nth term of a geometric sequence is given by the formula a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number. This formula allows direct calculation of any term without listing all previous terms.

Handling Negative and Fractional Common Ratios

When the common ratio is negative or a fraction, the terms alternate in sign or decrease in magnitude respectively. This affects the sequence's pattern and must be carefully applied in the formula to correctly find terms like a_40.