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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

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  • $\begingroup$ 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. $\endgroup$ May 20 '20 at 6:47
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$4, 25, 100$ 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.

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