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Let common ratio be $r$ and the first term be $a$. Then $ar^2\leq100 \implies r^2\leq10$

It's quite easy if we consider $r$ to be a natural number.

For $r=2$, $10\leq a\leq25$ and for $r=3$, $10\leq a\leq11$.

We get $18$ values. The problem arises when $r$ is not a natural number. $40,60,90$ definitely fulfils my condition, but I haven't been able to think of a foolproof method to find the number of GP's

Tatai
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1 Answers1

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Hint: The common ratio must be a rational number. Express it as a fraction $r=p/q$ in lowest terms, that is, such that $p$ and $q$ do not have any common factors.

Your geometric series of three is then $(a, ap/q, ap^2/q^2)$. They are all integers, so what do you know about $a$ and $q^2$? This should limit the possibilities of $q$ quite much. You still have to go through several cases, but this should help.

Jukka Kohonen
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  • I believe you are underestimating how much of a mess this is to count. – Rushabh Mehta Sep 02 '21 at 01:25
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    I'm not saying it is a one-liner, but you do not need to consider all $\binom{91}{3}=121485$ triples from ${10,11,\ldots,100}$. When $q$ is large, there are not many cases of $a$ to consider. When $q$ is small you can also use the fact that $p/q < \sqrt{10}$. Of couse, there is also the brute-force solution. – Jukka Kohonen Sep 02 '21 at 01:43
  • Those extras such as $40,60,90$ occur only when $a=kp^2$ for some integer $2≤p≤10$. That is not too many special cases to deal with by hand. – MJD Sep 02 '21 at 15:22
  • MJD: Quite true. If $r=p/q$ in lowest terms, then $\gcd(p^2,q^2)=1$, so $ap^2/q^2$ being an integer implies that $q^2$ divides $a$. (There may be other ways of showing the same result.) For example when $q=9$, you only need to try out a single value $a=1 \cdot 9^2 = 81$ (any other multiples are too large). – Jukka Kohonen Sep 02 '21 at 15:29
  • Yes, I agree the approach is feasible. I was addressing Don Thousand's implication that it would be too hard to count. There are a lot of cases though. I think it might be wise to count $k=1$ and $k>1$ separately. – MJD Sep 02 '21 at 15:41