Answer :
Final answer:
In a recursively defined sequence, after computing the values, we determine that statements E (f(2)=5) and C (f(5)=33) are true, while the other statements provided are false.
Explanation:
The question asks to evaluate a recursively defined sequence to determine the truth value of several statements regarding the sequence. The sequence is defined as f(1) = 3, and f(n+1) = 2 · f(n) - 1. We can use these definitions to compute the values in the sequence and check the statements.
f(2) = 2 · f(1) - 1 = 2 · 3 - 1 = 5, so statement E is true.
f(3) = 2 · f(2) - 1 = 2 · 5 - 1 = 9, so statement A is false (The correct value is 9, not 10).
f(4) = 2 · f(3) - 1 = 2 · 9 - 1 = 17, so statement D is false (The correct value is 17, not 18).
f(5) = 2 · f(4) - 1 = 2 · 17 - 1 = 33, so statement C is true.
f(6) = 2 · f(5) - 1 = 2 · 33 - 1 = 65, so statement B is false (The correct value is 65, not 66).
The true statements about the sequence are that f(2) = 5 and f(5) = 33.
Answer:
- C) f(5)= 33
- E) f(2) = 5
Step-by-step explanation:
The recursive definition tells you that values of f(n) will all be odd. (They are one less than an even number.) This observation eliminates all choices except C and E.
The first few values of f(n) are ...
3, 5, 9, 17, 33, 65, ...
From this list, we can see that f(2) = 5 and f(5) = 33, matching choices C and E.
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The explicit formula is ...
[tex]f(n)=2^{n}-1[/tex]
