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 a₁₂ when a₁ = 5, r = - 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 \(a_1 = 5\), the common ratio \(r = -2\), and the term to find is the 12th term, so \(n = 12\).
3. Substitute the known values into the formula: \(a_{12} = 5 \times (-2)^{12-1}\).
4. Simplify the exponent expression: calculate the power \((-2)^{11}\), keeping in mind the sign and magnitude.
5. Multiply the first term by the result of the exponentiation to express \(a_{12}\) fully as \$5 \times (-2)^{11}$.

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. For example, if the first term is 5 and the ratio is -2, the sequence progresses as 5, -10, 20, and so on.

General Term Formula of a Geometric Sequence

The nth term of a geometric sequence can be found using 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.

Exponentiation and Negative Ratios

When the common ratio is negative, powers of the ratio alternate in sign depending on whether the exponent is even or odd. Understanding how to handle negative bases raised to integer powers is essential to correctly compute terms like a_12 in the sequence.