How many Geometric Progressions consisting $5$ terms are possible between $1$ to $300$?

Question

Eg: $2,4,8,16,32.$ or $3,9,27,91,243.$ The common ratio has to be $>1$. i.e. It has to be an increasing GP.

Answer 1

Hint: The first term of a geometric series are given as $$a,ar,ar^2,ar^3,ar^4,ar^5.$$ and the list of above numbers is increasing (even if $r=1$), so all boils down to finding all solutions of $$ar^5\leq 300.$$ Start by considering for which $r$ it can hold, then focus on $a$.

Answer 2

Let your sequence be $a,ar,ar^2,ar^3,ar^4$. You want $r$ rational, $a$ an integer, $ar^4$ an integer, and $ar^4\leq 300$.

Now if $r=\frac{p}{q}$ is in reduced form, then $ar^4=\frac{ap^4}{q^4}$. Since $q$ and $p$ are relatively prime, this means that $q^4$ divides $a$, so you have that $ar^4=\frac{a}{q^4}p^4\geq p^4$.

So we know that $p^4<300$. That means that $p=2,3,4$. (Since $r>1$, we know that $p\neq 1$.)

This gives us a list of possible $r=\frac{p}{q}$:

$$\frac{4}{1},\frac{4}{3},\frac{3}{1},\frac{3}{2},\frac{2}{1}$$

For each of these, you need to count all $d$ such that $dp^4\leq 300$ (since then we can use $a=dq^4$.)