Which of the following functions represents a geometric sequence? Why?
Answer Choices
The function f(x)=x7 represents a geometric sequence because each term is raised to the seventh power to make it greater.
The function f(x)=7x represents a geometric sequence because each term is the next higher multiple of 7.
The function f(x)=7x represents a geometric sequence because each term is 7 times as great as the previous term.
The function f(x)=−7x+3 represents a geometric sequence because each term is 7 less than the previous term.
check the ratio between terms. If it is constant, you have a GP.
For example, check the first 3 terms of
f(x) = -7x+3
f(1) = -4
f(2) = -11
f(3) = -18
(-11)/(-4) ≠ (-18)/(-11) so not a GP
Use ^ for exponents, as in
x^7 and 7^x
connexus students 2018 unit 2 lesson 5
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A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant factor. To determine which of the given functions represents a geometric sequence, we need to check if each term is related to the previous term by multiplication or division.
Let's analyze each function:
1. The function f(x)=x^7: This function does not represent a geometric sequence because each term is raised to the seventh power, not multiplied by a constant factor.
2. The function f(x)=7x: This function represents a geometric sequence because each term is the next higher multiple of 7. Each term can be obtained by multiplying the previous term by 7.
3. The function f(x)=7x: This function also represents a geometric sequence. Each term is 7 times as great as the previous term, indicating a constant multiplication factor.
4. The function f(x)=-7x+3: This function does not represent a geometric sequence because each term is obtained by subtracting 7 from the previous term, not multiplying it by a constant factor.
Therefore, the correct answer is: The function f(x) = 7x represents a geometric sequence because each term is 7 times as great as the previous term.
To determine which function represents a geometric sequence, we need to understand the characteristics of a geometric sequence. A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Let's analyze each function to see if it matches the characteristics of a geometric sequence:
1. The function f(x) = x^7 does not represent a geometric sequence. Taking the seventh power of each term does not result in a fixed, non-zero common ratio. Hence, this function is not a geometric sequence.
2. The function f(x) = 7x does represent a geometric sequence. Each term is the next higher multiple of 7, indicating a common ratio of 7. For example, if x = 1, f(1) = 7, and if x = 2, f(2) = 14 = 7 * 2. The common ratio is 7, and each term is obtained by multiplying the previous term by the common ratio.
3. The function f(x) = 7x also represents a geometric sequence. As mentioned in the previous option, each term is obtained by multiplying the previous term by 7. This function, therefore, also has a fixed, non-zero common ratio, which confirms it as a geometric sequence.
4. The function f(x) = -7x + 3 does not represent a geometric sequence. The difference between consecutive terms is not a fixed multiple, indicating that it does not follow the definition of a geometric sequence.
Based on the analysis, options 2 and 3 both represent geometric sequences because they have a fixed, non-zero common ratio obtained by multiplying the previous term by a constant.