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, ...

Verified step by step guidance

1. Identify the first term \( a_1 \) of the geometric sequence. In this sequence, the first term is \( 3 \).
2. Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{-6}{3} \).
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.
4. Substitute the values \( a_1 = 3 \), \( r \) (from step 2), and \( n = 11 \) into the formula: \[ S_{11} = 3 \times \frac{1 - r^{11}}{1 - r} \].
5. 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.

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, 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.