The two geometric means between numbers 1 and 64 are _________

Answer

To find the two geometric means between the numbers 1 and 64, we can follow these steps:

### Step 1: Understand the Structure of the Geometric Progression (GP)
In a geometric progression, each term after the first is found by multiplying the previous term by a constant called the common ratio (r). If we have two geometric means between 1 and 64, we can denote the sequence as:
- First term (a1) = 1
- Second term (G1) = ?
- Third term (G2) = ?
- Fourth term (a4) = 64

### Step 2: Set Up the Terms
We can express the terms in terms of the first term (a) and the common ratio (r):
- a1 = a = 1
- a2 = ar = G1
- a3 = ar^2 = G2
- a4 = ar^3 = 64

### Step 3: Substitute Known Values
Since we know a1 = 1, we can substitute this into the equation for a4:
- ar^3 = 64
- 1 * r^3 = 64
- r^3 = 64

### Step 4: Solve for the Common Ratio (r)
To find r, we take the cube root of both sides:
- r = 64^(1/3)
- r = 4

### Step 5: Find the Geometric Means
Now that we have the common ratio (r = 4), we can find the geometric means:
- G1 = ar = 1 * 4 = 4
- G2 = ar^2 = 1 * 4^2 = 1 * 16 = 16

### Final Answer
The two geometric means between the numbers 1 and 64 are **4 and 16**.

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