find the number of geometric progressions that has three terms and lie in the range 1 to 100 inclusive of both numbers such that all the numbers are distinct and the common ratio is a natural number.

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Akshay-Since the progression has only 3 terms and it is geometric and it must include 1 and 100, then the following statements must be true:1. The progression is 1, x, 1002. x is a natural number3. The common ratio is x/1 = 100/xSolving statement 3:x/1 = x = 100/xx^2=100x=10 (Note: -10 is invalid because it is not a natural number)So, the progression is:1, 10, 100common ratio=10Hope that helped