Find 5 G.M. between 576 and 9.

Answer

To find 5 geometric means (G.M.) between 576 and 9, we can follow these steps:

### Step 1: Identify the first term (a) and the last term (b)
Let the first term \( a = 9 \) and the last term \( b = 576 \).

### Step 2: Determine the number of terms
Since we need to find 5 geometric means between 9 and 576, the total number of terms will be \( n = 5 + 2 = 7 \) (including the two endpoints).

### Step 3: Use the formula for the nth term of a geometric progression
The nth term of a geometric progression can be expressed as:
\[ b = a \cdot r^{n-1} \]
where \( r \) is the common ratio.

### Step 4: Substitute the known values into the formula
Substituting the known values into the formula gives:
\[ 576 = 9 \cdot r^{6} \]

### Step 5: Solve for \( r^{6} \)
To isolate \( r^{6} \), divide both sides by 9:
\[ r^{6} = \frac{576}{9} \]
Calculating the right side:
\[ r^{6} = 64 \]

### Step 6: Solve for \( r \)
Now, take the sixth root of both sides:
\[ r = 64^{\frac{1}{6}} = 2 \]
(Note: We can also consider \( r = -2 \) for negative terms.)

### Step 7: Find the 5 geometric means
Now we can find the 5 geometric means using the common ratio \( r \):
- First G.M.: \( 9 \cdot r = 9 \cdot 2 = 18 \)
- Second G.M.: \( 18 \cdot r = 18 \cdot 2 = 36 \)
- Third G.M.: \( 36 \cdot r = 36 \cdot 2 = 72 \)
- Fourth G.M.: \( 72 \cdot r = 72 \cdot 2 = 144 \)
- Fifth G.M.: \( 144 \cdot r = 144 \cdot 2 = 288 \)

### Step 8: List the results
The 5 geometric means between 9 and 576 are:
- 18, 36, 72, 144, 288