If the product of 3 consecutive terms of G>P. is 27 , the find the middle term.

Answer

To solve the problem, we need to find the middle term of three consecutive terms in a geometric progression (GP) whose product is given as 27. Let's break down the solution step by step.

### Step 1: Define the terms of the GP
Let the three consecutive terms of the GP be:
- First term: \( \frac{a}{r} \)
- Second term: \( a \)
- Third term: \( ar \)

### Step 2: Write the equation for the product
According to the problem, the product of these three terms is equal to 27. Therefore, we can write:
\[ \left(\frac{a}{r}\right) \cdot a \cdot (ar) = 27 \]

### Step 3: Simplify the product
Now, let's simplify the left-hand side:
\[ \frac{a}{r} \cdot a \cdot ar = \frac{a^3 r}{r} = a^3 \]
So, we have:
\[ a^3 = 27 \]

### Step 4: Solve for \( a \)
To find \( a \), we take the cube root of both sides:
\[ a = 27^{\frac{1}{3}} = 3 \]

### Step 5: Identify the middle term
In the GP, the middle term is simply \( a \). Therefore, the middle term is:
\[ \text{Middle term} = a = 3 \]

### Final Answer
The middle term of the three consecutive terms of the GP is \( 3 \).

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