# Geometric sequence not correct?

So I was checking some Khan Academy excercise about a sequence and it went something like this...

$$4, 25, 100...$$

It said that $$f(1)=4$$, $$f(2)=25$$ and $$f(3)=f(1)f(2).$$

So I was thinking about how to create an implicit formula for that, but I couldn't do it, instead I came up with a recursive formula.

$$f(n)= 4$$ , if $$n = 1.$$

$$f(n)=\frac{25}{16}4^n$$ if $$n>1$$.

Is this ok? Because I'm not exactly getting $$4$$ if I use that formula, instead I get a $$6.25$$ and therefore I was not able to create an implicit formula.

I also tried $$4^n+9x$$ for $$x=0,1,4,16,64...$$ when $$n=1,2,3,4...$$ that one works fine, the only problem it's that I don't know how to make $$x=0$$, I was thinking about this sequence, but doesn't seem to work well when I put it in Desmos... https://oeis.org/search?q=0%2C1%2C4%2C16%2C64%2C256&sort=&language=&go=Search

Any ideas?

Thanks! :3

• If you have $a_{n+2}=a_{n+1}a_n$ then consider $b_n=\log{(a_n)}$. For this sequence we have the recurrence $b_{n+2}=b_{n+1}+b_n$ which is now a standard form linear recurrence. May 20 '20 at 6:47

$$4, 25, 100$$ is not a geometric sequence. The formula of the $$n$$-th term of a geometric sequence is the following:$$a_n = a_{n-1}\times r$$, where $$r$$ is the factor between the terms (called the common ratio). So you can easily check that your numbers are not consecutive terms of a geometric sequence.