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.

Question

5 or 33 or 18 or 10 or 66

Answer 1

Given f(1) = 3.

f(n+1) = 2f(n) - 1.

Let n+1 = 2.

f(2) = 2f(1) - 1, but f(1) = 3 so f(2) = 2*3 - 1 = 5.

Now let n+1 =3.

f(3) = 2f(2) - 1, but f(2) = 5 so f(3) = 2*5 - 1 = 9.

From these examples it is obvious that f(n+1) will always be odd, so 18, 10 and 66 as solutions won't work.

It has been shown that f(2) = 5, so that is a solution.

Notice that f(1) = 3, f(2) = f(1) + 2, f(3) = f(2) + 4.

The pattern appears to be f(n+1) = f(n) + 2ⁿ

Let n+1 = 4. f(4) = f(3) + 2³, but f(3) = 9 (see above), so f(4) = 9+8 = 17.

If you try n+1 = 5, then you'll see that f(5) = f(4) + 2⁴ = 17+16 = 33, so 33 is also a solution.

Answer 2

You have not asked a question, I think!

However, the general solution of the difference equation is f(x)=2^x + 1. That should help.