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

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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.

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 $n = 40$.

Substitute the known values into the formula: $a_{40} = 1000 \times \left(-\frac{1}{2}\right)^{40-1}$.

Simplify the exponent expression: $a_{40} = 1000 \times \left(-\frac{1}{2}\right)^{39}$.

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 $a_{40}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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