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(5)=33
f(3)=10
f(4)=18
f(6)=66
f(2)=5
I cannot seem to figure out what to do, I even looked at my readings I do NOT want the answers I just need help on how to figure them out. That's why I didn't put any answer to what I think it may be, because I don't know how to solve this.
To find the values of the sequence, you can follow the recursive definition and calculate each term step by step.
Let's start with calculating f(2) based on the given recursive rule:
f(2) = 2 * f(1) - 1
= 2 * 3 - 1
= 6 - 1
= 5
Next, we can calculate f(3) using the same rule:
f(3) = 2 * f(2) - 1
= 2 * 5 - 1
= 10 - 1
= 9 (not 10)
Now, let's find f(4):
f(4) = 2 * f(3) - 1
= 2 * 9 - 1
= 18 - 1
= 17 (not 18)
Continuing, we can calculate f(5):
f(5) = 2 * f(4) - 1
= 2 * 17 - 1
= 34 - 1
= 33
Lastly, we will find f(6):
f(6) = 2 * f(5) - 1
= 2 * 33 - 1
= 66 - 1
= 65 (not 66)
Therefore, the correct statements about the sequence are:
- f(5) = 33
- f(3) = 9
- f(4) = 17
- f(6) = 65
- f(2) = 5
To find the values of the sequence defined recursively, you need to apply the recursive rule repeatedly, starting from the initial value given.
To solve this specific sequence, let's calculate each value step by step:
1. f(1) = 3 (given)
2. f(2) = 2f(1) - 1
Plugging in the value of f(1): f(2) = 2(3) - 1 = 6 - 1 = 5
3. f(3) = 2f(2) - 1
Plugging in the value of f(2): f(3) = 2(5) - 1 = 10 - 1 = 9
4. f(4) = 2f(3) - 1
Plugging in the value of f(3): f(4) = 2(9) - 1 = 18 - 1 = 17
5. f(5) = 2f(4) - 1
Plugging in the value of f(4): f(5) = 2(17) - 1 = 34 - 1 = 33
6. f(6) = 2f(5) - 1
Plugging in the value of f(5): f(6) = 2(33) - 1 = 66 - 1 = 65
Now that we have calculated the values, we can compare them to the statements given:
- f(5) = 33 (true, as calculated)
- f(3) = 10 (false, it is 9)
- f(4) = 18 (false, it is 17)
- f(6) = 66 (false, it is 65)
- f(2) = 5 (true, as calculated)
So, the only true statements are f(5) = 33 and f(2) = 5.
Recursion means you are using the terms before it to build the new term : )
f(1)=3 that means the function at x=1 gives a y value of 3
f(n+1)=2(f(n)) - 1 *
So n=1 to begin
so
f(2) is f(n+1) where n=1
so sub in n=1 into the * equation
f(2) = f(1+1) = 2(f(1)) - 1
but we know that f(1) = 3 so where we see f(1) replace it with 3
f(2) = f(1+1) = 2(3) - 1
f(2) = 6-1
f(2) = 5
so f(2) matches the one in the list given so f(2)=5 is correct.
Now try f(n+1)=2(f(n)) - 1 *
where we are looking for f(3), so that is n+1 = 3 so n=2
f(2+1) = 2(f(2)) - 1
now since we just found f(2)=5 we can use that information for f(3)
f(3) = 2(5) - 1
f(3) = 10 - 1
f(3) = 9 the list said f(3) was 10, so that one does not match since f(3)=9
We use the function at the previous step to find the next step : )