The first term of an A.P is 7 , the last term is 47 and the sum is 432 . Find the number of terms and the common difference .

Answer

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

### Given:
- First term (a) = 7
- Last term (Tn) = 47
- Sum of n terms (S) = 432

### Step 1: Use the formula for the sum of the first n terms of an A.P.
The formula for the sum of the first n terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (a + T_n) \]
Where:
- \( S_n \) = sum of the first n terms
- \( n \) = number of terms
- \( a \) = first term
- \( T_n \) = last term

### Step 2: Substitute the known values into the formula.
Substituting the values we have:
\[ 432 = \frac{n}{2} \times (7 + 47) \]
\[ 432 = \frac{n}{2} \times 54 \]

### Step 3: Simplify the equation.
Multiply both sides by 2 to eliminate the fraction:
\[ 864 = n \times 54 \]

### Step 4: Solve for n.
Now, divide both sides by 54:
\[ n = \frac{864}{54} \]
\[ n = 16 \]

### Step 5: Find the common difference (d).
We know that the nth term (Tn) can also be expressed as:
\[ T_n = a + (n - 1) \times d \]
Substituting the known values:
\[ 47 = 7 + (16 - 1) \times d \]
\[ 47 = 7 + 15d \]

### Step 6: Isolate d.
Subtract 7 from both sides:
\[ 40 = 15d \]
Now divide both sides by 15:
\[ d = \frac{40}{15} \]
\[ d = \frac{8}{3} \]

### Final Results:
- Number of terms (n) = 16
- Common difference (d) = \( \frac{8}{3} \)

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