Find two geometric means between 5 and 135

We are effectively looking for a and b such that 5, a, b, 135 is a geometric sequence.

This sequence has common ratio 3 √ 1355 = 3, hence a=15 and b=45

Explanation:

In a geometric sequence, each intermediate term is the geometric mean of the term before it and the term after it.

So we want to find a and b such that 5, a, b, 135 is a geometric sequence.

If the common ratio is r then:

a=5r b=ar=5 r2 135=br=5 r3

Hence r3 = 1355 = 27, so r = 3 √27 = 3

Then a=5r=15 and b=ar=15⋅3=45