In Exercises 9–16, 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: $a{n}_{}=a{1}_{}\cdot r^{n-1}$, where $a{1}_{}$ is the first term, $r$ is the common ratio, and $n$ is the term number.
2. Identify the given values: $a{1}_{}=\; 1000$, $r=\; -\backslash frac\{1\}\{2\}$, and $n=\; 40$.
3. Substitute the values into the formula: $a{40}_{}=\; 1000\cdot \backslash left(-\backslash frac\{1\}\{2\}\backslash right)^\{40\; -\; 1\}$.
4. Simplify the exponent expression: $40\; -\; 1\; =\; 39$, so the formula becomes $a{40}_{}=\; 1000\cdot \backslash left(-\backslash frac\{1\}\{2\}\backslash right)^\{39\}$.
5. Evaluate the power $\backslash left(-\backslash frac\{1\}\{2\}\backslash right)^\{39\}$ carefully, noting that since the exponent 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.

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.

Evaluating Powers of Negative Fractions

When the common ratio is a negative fraction, raising it to a power involves considering both the magnitude and the sign. Even powers make the term positive, while odd powers keep it negative, and fractional bases require careful exponentiation.