Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a₇ when a₁ = 2, r = 3

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: \(a_1 = 2\), \(r = 3\), and \(n = 7\) (since we want to find \(a_7\)).
3. Substitute the known values into the formula: \(a_7 = 2 \times 3^{7-1}\).
4. Simplify the exponent: \$7 - 1 = 6\(, so the expression becomes \)a_7 = 2 \times 3^6$.
5. Calculate \$3^6\( and then multiply by 2 to find the value of \)a_7$.

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 2 and the ratio is 3, the sequence is 2, 6, 18, 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 3^(7-1) when finding the 7th term.