If the sum of 4 terms of an A.P. is 36, then find the A.M of the A.P.

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Understand the Problem
We are given that the sum of 4 terms of an Arithmetic Progression (A.P.) is 36. We need to find the Arithmetic Mean (A.M.) of these terms.

### Step 2: Represent the 4 Terms of the A.P.
Let the four terms of the A.P. be represented as:
- First term: \( a - 3d \)
- Second term: \( a - d \)
- Third term: \( a + d \)
- Fourth term: \( a + 3d \)

Here, \( a \) is the middle term, and \( d \) is the common difference.

### Step 3: Write the Equation for the Sum of the Terms
The sum of these four terms can be expressed as:
\[ (a - 3d) + (a - d) + (a + d) + (a + 3d) = 36 \]

### Step 4: Simplify the Equation
Now, combine like terms:
\[ a - 3d + a - d + a + d + a + 3d = 36 \]
This simplifies to:
\[ 4a = 36 \]

### Step 5: Solve for \( a \)
Now, divide both sides by 4 to find \( a \):
\[ a = \frac{36}{4} = 9 \]

### Step 6: Find the Arithmetic Mean (A.M.)
The Arithmetic Mean of the four terms is given by the formula:
\[ \text{A.M.} = \frac{\text{Sum of the terms}}{\text{Number of terms}} = \frac{(a - 3d) + (a - d) + (a + d) + (a + 3d)}{4} \]
From our earlier simplification, we know that:
\[ \text{A.M.} = \frac{4a}{4} = a \]
Since we found \( a = 9 \):
\[ \text{A.M.} = 9 \]

### Final Answer
The Arithmetic Mean of the A.P. is **9**.

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