High School

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Is the given number a term in both the sequence [tex]f(n) = f(n-1) + 5[/tex] and the sequence [tex]f(n) = 3(2)^{n-1}[/tex], if [tex]f(1) = 3[/tex]?

A. 8
B. 18
C. 24
D. 48

Answer :

Final answer:

The number 24 is the only given option that is a term in both sequences, when we evaluate the sequences using their respective formulation starting with f(1) = 3. So, the correct answer is c) 24.

Explanation:

To determine if the given number is a term in both the sequences f(n) = f(n-1) + 5 and f(n) = 3(2)¹⁽¹, with the initial condition f(1) = 3, we must first generate terms from both sequences to check for common values. Starting with f(1) = 3, for the first sequence, the subsequent terms would be f(2) = 8, f(3) = 13, and so on, increasing by 5 each time. For the second sequence, f(2) = 3×2, f(3) = 3×4, and so forth, doubling the multiplier of 3 each time.

Now, we consider the provided options and check for their presence in both sequences:

  • a. 8 - Present in the first sequence as f(2), but not in the second sequence.
  • b. 18 - Not present as obtained from initial 5 term calculations for both sequences.
  • c. 24 - Present as f(4) in the first sequence and also present in the second sequence.
  • d. 48 - Not present in the initial 5 term calculations for both sequences.

Based on the search, 24 (option c) is a term in both sequences.

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