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 a₈ when a₁ = 6, r = 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 $a_1 = 6$, the common ratio $r = 2$, and the term to find is the 8th term, so $n = 8$.

Substitute the known values into the formula: $a_8 = 6 \times 2^{8-1}$.

Simplify the exponent: $8 - 1 = 7$, so the expression becomes $a_8 = 6 \times 2^7$.

Evaluate \$2^7$ (which is \(2$ multiplied by itself 7 times) and then multiply the result by 6 to find \)a_8$.

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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. For example, if the first term is 6 and the ratio is 2, the sequence is 6, 12, 24, 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 in Sequences

Exponentiation is used in the general term formula to raise the common ratio to the power of (n-1). Understanding how to compute powers is essential to correctly find terms in geometric sequences, such as calculating 2^(7) when finding the 8th term.

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