Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 11 terms of the geometric sequence: 3, - 6, 12, - 24, ...

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Identify the first term \( a_1 \) of the geometric sequence. In this sequence, the first term is \( 3 \).

Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{-6}{3} \).

Recall the formula for the sum of the first \( n \) terms of a geometric sequence: \[ S_n = a_1 \times \frac{1 - r^n}{1 - r} \] where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Substitute the values \( a_1 = 3 \), \( r \) (from step 2), and \( n = 11 \) into the formula: \[ S_{11} = 3 \times \frac{1 - r^{11}}{1 - r} \].

Simplify the expression by calculating \( r^{11} \), then perform the subtraction and division inside the fraction, and finally multiply by 3 to find the sum of the first 11 terms.

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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, -6, 12, -24, ..., the common ratio is -2.

Sum of the First n Terms of a Geometric Sequence

The sum of the first n terms of a geometric sequence can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms. This formula applies when r is not equal to 1.

Substitution and Simplification

To find the sum, substitute the known values (a = 3, r = -2, n = 11) into the sum formula and simplify carefully. This involves calculating powers of the ratio and performing arithmetic operations to get the final sum.