I am very confused on how a recursive rule works.

Question

anonymous · Answer

The basic idea is that you start with a number, apply some rule, and you get a new number. You then apply the same rule to this new number to get yet another new number.

Here's an example. Say your rule is $a_n=2a_{n-1}$. In other words, you start with some number $a_1$, let's say $a_1=1$.

Then the next number, according to this rule, is twice the first number. That is, $a_2=2a_1$, so $a_2=2\times a_1=2\times1=2$. What would the next number $a_3$ be?

anonymous · Answer

4?

anonymous · Answer

Right, each successive number is double the previous one. Do you have a specific rule you're working with?

anonymous · Answer

f(1) = 65,536, f(n) = Square Root of f(n-1)

anonymous · Answer

I'm supposed to be finding the first 4 terms

anonymous · Answer

Okay, so the first term is $f(1)=65536$.

The next term, according to the rule, would be$$f(2)=\sqrt{f(1)}$$or$$f(2)=\sqrt{65536}=256$$(since $256^2=65536$).

The next term would be$$f(3)=\sqrt{f(2)}=\sqrt{256}=\cdots$$Does that make sense?

anonymous · Answer

So The square root of f(2) is equal to the square root of 256?

anonymous · Answer

Yes

anonymous · Answer

Note that $256$ is a square number, so you *can* reduce it further. (Hence the $\cdots$)

anonymous · Answer

So the first 4 terms would be 65,536, 256, 16, 4, right?

anonymous · Answer

Thank you for helping me out. I really appreciate it.

anonymous · Answer

That's correct. You're welcome!