In an AP, the first term is -4, the last term is 29 and the sum of all its terms is 150. Find its common difference.

Answer

To find the common difference of the arithmetic progression (AP) with the given parameters, we can follow these steps:

### Step 1: Identify the given values
- First term (a) = -4
- Last term (l) = 29
- Sum of all terms (S) = 150

### Step 2: Use the formula for the sum of an AP
The formula for the sum of the first n terms of an arithmetic progression is given by:
\[ S = \frac{n}{2} \times (a + l) \]
Where:
- S = sum of the terms
- n = number of terms
- a = first term
- l = last term

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula:
\[ 150 = \frac{n}{2} \times (-4 + 29) \]

### Step 4: Simplify the equation
Calculate \( -4 + 29 \):
\[ -4 + 29 = 25 \]
Now substitute this back into the equation:
\[ 150 = \frac{n}{2} \times 25 \]

### Step 5: Solve for n
Multiply both sides by 2 to eliminate the fraction:
\[ 300 = n \times 25 \]
Now divide both sides by 25:
\[ n = \frac{300}{25} = 12 \]

### Step 6: Use the formula for the nth term of an AP
The nth term of an AP can be expressed as:
\[ l = a + (n - 1) \times d \]
Where:
- d = common difference

### Step 7: Substitute known values to find d
Substituting the known values into the nth term formula:
\[ 29 = -4 + (12 - 1) \times d \]
This simplifies to:
\[ 29 = -4 + 11d \]

### Step 8: Solve for d
Add 4 to both sides:
\[ 29 + 4 = 11d \]
\[ 33 = 11d \]
Now divide both sides by 11:
\[ d = \frac{33}{11} = 3 \]

### Final Answer
The common difference \( d \) is 3.

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