A sequence is defined recursively by the following rules:
f(1)=3
f(n+1)=2⋅f(n)−1
Which of the following statements is true about the sequence? Select all that apply.
f(3)=10
f(2)=5
f(4)=18
f(6)=66
f(5)=33
well, just list the first few elements and check:
f(1) = 3
f(2) = 2*3-1 = 5
f(3) = 2*5-1 = 9
and so on
To determine which of the statements are true about the sequence, we can use the recursive definition given.
Let's start by finding the values of f(2), f(3), f(4), f(5), and f(6) step by step.
Given: f(1) = 3
Using the recursive rule:
f(2) = 2*f(1) - 1
= 2*3 - 1
= 6 - 1
= 5
f(3) = 2*f(2) - 1
= 2*5 - 1
= 10 - 1
= 9
f(4) = 2*f(3) - 1
= 2*9 - 1
= 18 - 1
= 17
f(5) = 2*f(4) - 1
= 2*17 - 1
= 34 - 1
= 33
f(6) = 2*f(5) - 1
= 2*33 - 1
= 66 - 1
= 65
Now we can determine which statements are true:
f(3) ≠ 10 (False),
f(2) = 5 (True),
f(4) ≠ 18 (False),
f(6) ≠ 66 (False),
f(5) = 33 (True).
Therefore, the correct statements are: f(2) = 5 and f(5) = 33.