Problem 12
Question
Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1, and common ratio, r. Find \(a_{12}\) when \(a_{1}=4, r=-2.\)
Step-by-Step Solution
Verified Answer
The 12th term of the geometric sequence is -4096.
1Step 1: Identify the given variables
Here, the first term (\(a_1\)) is 4, the common ratio (r) is -2, and we are asked to solve for the 12th term (\(a_{12}\)) of the sequence. Thus, \(a_1\) = 4, r = -2 and n = 12.
2Step 2: Plug in the variables into the geometric sequence formula
To calculate the 12th term of the sequence, apply the geometric sequence formula, substitute the identified variables into the formula: \(a_n = a_1 \times r^{(n-1)}\). Therefore, \(a_{12} = 4 \times (-2)^{11}\).
3Step 3: Calculate term
Calculate the expression \(4 \times (-2)^{11}\). Given that 11 is an odd number, the result of \(-2^{11}\) will be negative, hence the result will be -4096.
Key Concepts
Geometric Sequence FormulaCommon Ratio in SequencesNth Term of a Sequence
Geometric Sequence Formula
Understanding the formula for a geometric sequence is essential for calculating any term within the sequence. A geometric sequence is made up of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. The formula for finding the nth term \(a_n\) of a geometric sequence is given by:
\[ a_n = a_1 \times r^{(n-1)} \]
where \(a_1\) is the first term of the sequence and \(r\) is the common ratio. This compact formula provides a direct method to calculate any term in the sequence without needing to find all the preceding terms. To use this formula effectively, you first need to identify the initial term and the common ratio from the sequence, as was done in the given exercise.
\[ a_n = a_1 \times r^{(n-1)} \]
where \(a_1\) is the first term of the sequence and \(r\) is the common ratio. This compact formula provides a direct method to calculate any term in the sequence without needing to find all the preceding terms. To use this formula effectively, you first need to identify the initial term and the common ratio from the sequence, as was done in the given exercise.
Common Ratio in Sequences
The common ratio is a cornerstone concept in geometric sequences. It is the factor by which subsequent terms in the sequence are generated. Calculating the common ratio is done by dividing any term by the previous term, and it remains consistent throughout the sequence.
\[ r = \frac{a_{4}}{a_{3}} = \frac{-24}{12} = -2 \]
As observed in the exercise provided, the common ratio can be any real number, including negative numbers, which will result in the sequence alternating between positive and negative values.
Identifying the Common Ratio
In most problems, you'll either be given the common ratio directly, as in the exercise, or you'll calculate it by examining two consecutive terms. For instance, if the third term of a sequence is 12 and the fourth term is -24, the common ratio \( r \) would be calculated as:\[ r = \frac{a_{4}}{a_{3}} = \frac{-24}{12} = -2 \]
As observed in the exercise provided, the common ratio can be any real number, including negative numbers, which will result in the sequence alternating between positive and negative values.
Nth Term of a Sequence
The 'nth term' is a way of referring to any term in a sequence, particularly when discussing the general case. It's the term you're looking to find, where 'n' is its position in the sequence.
In the provided exercise, the aim was to find the 12th term (\(a_{12}\)), which requires us to plug our known values into the geometric sequence formula. The formula provided is powerful as it allows us to jump directly to any term without laboriously calculating each preceding term.
In the provided exercise, the aim was to find the 12th term (\(a_{12}\)), which requires us to plug our known values into the geometric sequence formula. The formula provided is powerful as it allows us to jump directly to any term without laboriously calculating each preceding term.
Practical Application
When solving for the nth term, keep in mind that sequences can be infinite, but any single term is finite and exact. You would usually perform the calculations as shown in the steps: substitute the known values into the geometric sequence formula and simplify the mathematical expression to find the desired term.Other exercises in this chapter
Problem 12
In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$3+4+5+\cdots+(n+2)=\frac{n(n+5)}{2}$$
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In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{n}=a_{n-1}-20, a_{1}=50$$
View solution Problem 13
A die is rolled. Find the probability of getting an odd number.
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Use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{7} C_{7} $$
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