The first term of an A.P. is 3, the last term is 83 and the sum of all its terms is 903. Find the number of terms and the common difference of the A.P.

Answer

To solve the problem step by step, we will use the formulas related to Arithmetic Progressions (A.P.).

### Given:
- First term (a) = 3
- Last term (l) = 83
- Sum of all terms (S_n) = 903

### Step 1: Use the formula for the sum of an A.P.
The formula for the sum of the first n terms of an A.P. is given by:

\[ S_n = \frac{n}{2} \times (a + l) \]

### Step 2: Substitute the known values into the formula.
Substituting the values we have:

\[ 903 = \frac{n}{2} \times (3 + 83) \]

### Step 3: Simplify the equation.
Calculate \(3 + 83\):

\[ 3 + 83 = 86 \]

Now substitute this back into the equation:

\[ 903 = \frac{n}{2} \times 86 \]

### Step 4: Multiply both sides by 2 to eliminate the fraction.
\[ 1806 = n \times 86 \]

### Step 5: Solve for n.
Now, divide both sides by 86:

\[ n = \frac{1806}{86} \]

Calculating this gives:

\[ n = 21 \]

### Step 6: Use the formula for the nth term of an A.P.
The formula for the nth term (l) of an A.P. is given by:

\[ l = a + (n - 1) \cdot d \]

### Step 7: Substitute the known values into the nth term formula.
Substituting the values we have:

\[ 83 = 3 + (21 - 1) \cdot d \]

### Step 8: Simplify the equation.
Calculate \(21 - 1\):

\[ 21 - 1 = 20 \]

Now substitute this back into the equation:

\[ 83 = 3 + 20d \]

### Step 9: Isolate d.
Subtract 3 from both sides:

\[ 80 = 20d \]

### Step 10: Solve for d.
Now divide both sides by 20:

\[ d = \frac{80}{20} = 4 \]

### Final Results:
- Number of terms (n) = 21
- Common difference (d) = 4