The first and last terms of an A.P. are - 7 and 233 and the sum of the A.P. is 9153. Find the number of terms in the A.P.:

Options

A. 83
B. 81
C. 49
D. 99

Answer

To find the number of terms in the arithmetic progression (A.P.), we can use the formula for the sum of the first n terms of an A.P.:

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

where:
- \( S_n \) is the sum of the first n terms,
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( l \) is the last term.

Given:
- First term \( a = -7 \)
- Last term \( l = 233 \)
- Sum \( S_n = 9153 \)

### Step 1: Substitute the known values into the formula.

We substitute \( a \), \( l \), and \( S_n \) into the formula:

\[ 9153 = \frac{n}{2} \times (-7 + 233) \]

### Step 2: Simplify the expression inside the parentheses.

Calculate \( -7 + 233 \):

\[ -7 + 233 = 226 \]

Now, substitute this back into the equation:

\[ 9153 = \frac{n}{2} \times 226 \]

### Step 3: Multiply both sides by 2 to eliminate the fraction.

\[ 2 \times 9153 = n \times 226 \]

Calculating \( 2 \times 9153 \):

\[ 18306 = n \times 226 \]

### Step 4: Solve for \( n \).

Now, divide both sides by 226 to find \( n \):

\[ n = \frac{18306}{226} \]

Calculating the division:

\[ n = 81 \]

### Conclusion:

The number of terms in the A.P. is \( n = 81 \).

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