Question

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.

Answer 1

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 : )