The first and the last terms of an A.P. are 4 and 81 respectively . If the common difference is 7, how many terms are there in the A.P. and what is their sum?

Answer

To solve the problem step by step, we will follow the mathematical principles of an Arithmetic Progression (A.P.).

### Step 1: Identify the given values
- First term (a) = 4
- Last term (l) = 81
- Common difference (d) = 7

### Step 2: Use the formula for the nth term of an A.P.
The nth term of an A.P. can be expressed as:
\[ l = a + (n - 1) \cdot d \]
Where:
- \( l \) is the last term
- \( a \) is the first term
- \( n \) is the number of terms
- \( d \) is the common difference

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula:
\[ 81 = 4 + (n - 1) \cdot 7 \]

### Step 4: Simplify the equation
First, isolate the term involving \( n \):
\[ 81 - 4 = (n - 1) \cdot 7 \]
\[ 77 = (n - 1) \cdot 7 \]

### Step 5: Solve for \( n - 1 \)
Now, divide both sides by 7:
\[ n - 1 = \frac{77}{7} \]
\[ n - 1 = 11 \]

### Step 6: Solve for \( n \)
Now, add 1 to both sides to find \( n \):
\[ n = 11 + 1 \]
\[ n = 12 \]

### Step 7: Calculate the sum of the A.P.
The sum \( S_n \) of the first \( n \) terms of an A.P. can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]

### Step 8: Substitute the values to find the sum
Substituting the values we have:
\[ S_{12} = \frac{12}{2} \cdot (4 + 81) \]
\[ S_{12} = 6 \cdot 85 \]
\[ S_{12} = 510 \]

### Final Answers
- The number of terms \( n = 12 \)
- The sum of the A.P. \( S_n = 510 \)