Textbook Question
Evaluate each expression.
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 9, Problem 25
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 12 terms of the geometric sequence: 2, 6, 18, 54, ...
Verified step by step guidance1
Identify the first term \( a_1 \) of the geometric sequence. In this sequence, the first term is \( 2 \).
Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{6}{2} \).
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 \( n \) is the number of terms.
Substitute the values \( a_1 = 2 \), \( r \) (from step 2), and \( n = 12 \) into the formula to set up the expression for \( S_{12} \).
Simplify the expression by calculating \( r^{12} \), then compute the numerator and denominator separately before dividing to find the sum of the first 12 terms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 2, 6, 18, 54, the common ratio is 3 because each term is three times the previous one.
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Geometric Sequences - Recursive Formula
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 helps find the total sum without adding each term individually.
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Exponents and Powers
Exponents represent repeated multiplication of a base number. In the sum formula, r^n means multiplying the common ratio r by itself n times. Understanding how to work with exponents is essential for correctly applying the sum formula and simplifying expressions.
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Related Practice
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Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
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Evaluate each factorial expression. 16!/2!14!
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