Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a₇, the seventh term of the sequence. 3, 12, 48, 192, ...

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Identify the first term of the geometric sequence, which is \(a_1 = 3\).

Determine the common ratio \(r\) by dividing the second term by the first term: \(r = \frac{12}{3}\).

Write the formula for the general term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\).

Substitute the values of \(a_1\) and \(r\) into the formula to get the explicit formula for \(a_n\).

Use the formula to find the seventh term by substituting \(n = 7\) into \(a_n = a_1 \times r^{n-1}\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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, in the sequence 3, 12, 48, 192, ..., each term is multiplied by 4 to get the next term.

General Term Formula of a Geometric Sequence

The general term (nth term) of a geometric sequence is given by 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 you to find any term in the sequence without listing all previous terms.

Evaluating the nth Term

Once the general term formula is established, you substitute the desired term number (n) into the formula to calculate that specific term. For example, to find the 7th term, plug n = 7 into the formula and simplify to get a₇.

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for an to find a7, the seventh term of the sequence. 3, 12, 48, 192, ...