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![Margaret is writing a recursive function for the geometric sequence:
10,30,90,270,dots
She comes up with:
{[t(0)=10],[t(n)=t(n-1)*3]:}
What domain should Margaret use for
t so it generates the sequence?
Choose 1 answer:
(A)
n >= 0 where
n is an integer
(B)
n >= 0 where
n is any number
(C)
n >= 1 where
n is an integer
(D)
n >= 1 where
n is any number](https://externalquestionsimg-a4fuc2b6e0gtdqge.z01.azurefd.net/external-question-images/images/images/ka/Algebra-1/Sequences/General%20sequences/Q-33.png)
Margaret is writing a recursive function for the geometric sequence:She comes up with:What domain should Margaret use for so it generates the sequence?Choose answer:(A) where is an integer(B) where is any number(C) where is an integer(D) where is any number
Full solution
Q. Margaret is writing a recursive function for the geometric sequence:She comes up with:What domain should Margaret use for so it generates the sequence?Choose answer:(A) where is an integer(B) where is any number(C) where is an integer(D) where is any number
- Geometric sequence definition: Margaret is writing a recursive function for a geometric sequence. The first term of the sequence is given as , and each subsequent term is obtained by multiplying the previous term by . We need to determine the appropriate domain for the function that generates this sequence.
- Recursive function for the sequence: The sequence starts with and follows the pattern where each term is three times the previous term. This is a geometric sequence with a common ratio of . The recursive function provided is , with . The domain of the function will determine the valid input values for .
- Determining the domain of the function: Since the sequence starts with the first term , the domain must include . This eliminates options (C) and (D), which start with . Now we need to decide between being any number or being an integer.
- Including the first term: A sequence is defined by its terms at specific positions, which are discrete points. Therefore, the values of should be integers to correspond to the positions in the sequence. This means that cannot be any number; it must be an integer to make sense in the context of a sequence.
- Choosing the appropriate values for : Given that the sequence starts at and must be an integer to correspond to the discrete positions in the sequence, the correct domain for the function is where is an integer. This corresponds to option .
