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Magazine Chapter 8: Problem 3
Find the first four terms of the sequence. \(a_{n}=4(-2)^{n-1}\)
Short Answer
Expert verified
The first four terms are 4, -8, 16, and -32.
Step by step solution
01
Substitute n=1 into the formula
To find the first term of the sequence, substitute \(n=1\) into the formula \(a_n=4(-2)^{n-1}\). This gives us \(a_1 = 4(-2)^{1-1} = 4(-2)^0\). Since \((-2)^0 = 1\), \(a_1 = 4\times1 = 4\).
02
Substitute n=2 into the formula
To find the second term of the sequence, substitute \(n=2\) into the formula \(a_n=4(-2)^{n-1}\). This gives us \(a_2 = 4(-2)^{2-1} = 4(-2)^1\). Since \((-2)^1 = -2\), \(a_2 = 4\times(-2) = -8\).
03
Substitute n=3 into the formula
To find the third term of the sequence, substitute \(n=3\) into the formula \(a_n=4(-2)^{n-1}\). This gives us \(a_3 = 4(-2)^{3-1} = 4(-2)^2\). Since \((-2)^2 = 4\), \(a_3 = 4\times4 = 16\).
04
Substitute n=4 into the formula
To find the fourth term of the sequence, substitute \(n=4\) into the formula \(a_n=4(-2)^{n-1}\). This gives us \(a_4 = 4(-2)^{4-1} = 4(-2)^3\). Since \((-2)^3 = -8\), \(a_4 = 4\times(-8) = -32\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Geometric Sequences
Geometric sequences are a type of mathematical sequence where each term is generated by multiplying the previous term by a constant factor known as the "common ratio." These sequences have a clear pattern, making them easy to analyze and predict. In a geometric sequence, the ratio between any two consecutive terms is constant, which is the defining feature of this pattern.
- The general formula for a geometric sequence is usually given as \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio.
- In the provided example, the sequence formula given is \(a_n = 4(-2)^{n-1}\). Here, \(4\) is the coefficient that scales the term, and \(-2\) is the common ratio.
- Geometric sequences can alternate between negative and positive if the common ratio itself is negative, as seen in this case.
Exponents
Exponents are a mathematical notation indicating the number of times a number is to be multiplied by itself. In our sequence formula, exponents play a crucial role in determining the value of each term.
- In the formula \(a_n = 4(-2)^{n-1}\), the part \((-2)^{n-1}\) uses the concept of exponents where \(-2\) is the base and \(n-1\) is the exponent.
- Raising a number to the power of zero, such as \((-2)^0\), always results in one. This is why the first term \(a_1\) simplifies to \(4\times1=4\).
- The sign of the exponent determines the direction of the sequence when the base is negative. Odd exponents result in a negative number, while even exponents give a positive one.
Term Calculation
Calculating specific terms in a sequence involves substituting values into the formula to determine each subsequent number. This method streamlines finding terms without manually computing the whole sequence.
- The first step in calculating a term is to substitute the desired term's position number into \(n\) in the formula. For instance, the first term is calculated using \(n=1\).
- Once the substitution is complete, simplify the expression by computing the exponent term \((-2)^{n-1}\) and then multiplying by the coefficient, \(4\).
- Repeat these steps for each subsequent term until the required term is found. The calculated values for the first four terms in this sequence become \(4, -8, 16, -32\).
