Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a₅ when a₁ = -3, 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: \(a_1 = -3\), \(r = 2\), and \(n = 5\) (since we want to find \(a_5\)).

Substitute the known values into the formula: \(a_5 = -3 \times 2^{5-1}\).

Simplify the exponent: \(2^{5-1} = 2^4\).

Express the term \(a_5\) as \(a_5 = -3 \times 2^4\) and leave it in this form to find the exact value.

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주요 개념

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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 a₁ and the ratio is r, the sequence progresses as a₁, a₁r, a₁r², and so on.

General Term Formula of a Geometric Sequence

The nth term of a geometric sequence can be found using the formula aₙ = a₁ * r^(n-1), where a₁ 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.

Substitution and Evaluation

To find a specific term like a₅, substitute the given values of a₁, r, and n into the general term formula. Then, perform the exponentiation and multiplication carefully to evaluate the term accurately.

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a5 when a1 = -3, r = 2