The tw geometric means between the numbers 1 and 64 are

Options

A. 1 and 64
B. 4 and 16
C. 2 and 16
D. 8 and 16

Answer

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

### Step 1: Understand the geometric progression (GP)
In a geometric progression, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).

### Step 2: Identify the first and last terms
Here, the first term (a) is 1, and the last term (l) is 64. We need to find the two geometric means (G1 and G2) between these two numbers.

### Step 3: Determine the number of terms
In this case, we have:
- First term (a) = 1
- Second term (G1)
- Third term (G2)
- Fourth term (l) = 64

This means we have a total of 4 terms in the geometric progression.

### Step 4: Use the formula for the nth term of a GP
The nth term of a geometric progression can be expressed as:
\[ l = a \cdot r^{(n-1)} \]
where:
- \( n \) is the total number of terms,
- \( a \) is the first term,
- \( r \) is the common ratio.

In our case, we have:
\[ 64 = 1 \cdot r^{(4-1)} \]
\[ 64 = r^3 \]

### Step 5: Solve for the common ratio (r)
To find \( r \), we take the cube root of both sides:
\[ r = 64^{1/3} \]
\[ r = 4 \]

### Step 6: Calculate the geometric means
Now that we have the common ratio \( r \), we can find the geometric means:
- The first geometric mean (G1) is given by:
\[ G1 = a \cdot r = 1 \cdot 4 = 4 \]

- The second geometric mean (G2) is given by:
\[ G2 = a \cdot r^2 = 1 \cdot 4^2 = 1 \cdot 16 = 16 \]

### Step 7: List the complete series
The complete series of the geometric progression is:
1, 4, 16, 64

### Conclusion
The two geometric means between 1 and 64 are **4 and 16**.

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