Question

If f(1)=1f(1)=1 and f(n)=f(n-1)^(2)-1f(n)=f(n-1)
2
-1 then find the value of f(4)f(4).

Answer 1

f(4) = 0

Explanation:

For many sequences that involve recursive formulas, the quickest way to find the value of a given term is to evaluate those leading up to it. That seems to be the case here.

given

• f(1) = 1
• f(n) = f(n -1)² -1

find

• f(4)

solution

The next term after the one given is ...

f(2) = f(1)² -1 = 1² -1 = 0

Then we have ...

f(3) = f(2)² -1 = 0² -1 = -1

f(4) = f(3)² -1 = (-1)² -1 = 0

The value of f(4) is 0.

_____

Additional comment

Now that we have seen a few terms of the sequence, we can write an explicit formula for the terms after the first:

f(n) = (-1 +(-1)^n)/2 . . . integer n ≥ 2