Read the following text and answer the following questions on the basis of the same :
Amit has a packet of Candies .It consists of 288 candies .He arranges the candies in a way that first row contains 3 candies ,second row has 5 and third row has 7 and so on .
How many rows are there ?

Options

A. 10
B. 15
C. 16
D. 18

Answer

To solve the problem step by step, we will follow the arithmetic progression (AP) concept and the formula for the sum of an AP.

### Step-by-Step Solution:

1. **Identify the Pattern**:
The number of candies in each row forms an arithmetic progression:
- First row: 3 candies
- Second row: 5 candies
- Third row: 7 candies
- The pattern continues with an increase of 2 candies for each subsequent row.

2. **Define the First Term and Common Difference**:
- First term (a) = 3
- Common difference (d) = 5 - 3 = 2

3. **Sum of the First n Terms of an AP**:
The sum \( S_n \) of the first n terms of an arithmetic progression can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Here, we know that the total number of candies is 288, so:
\[ S_n = 288 \]

4. **Substituting Known Values**:
Substitute \( a = 3 \) and \( d = 2 \) into the sum formula:
\[ 288 = \frac{n}{2} \times (2 \times 3 + (n - 1) \times 2) \]
Simplifying this gives:
\[ 288 = \frac{n}{2} \times (6 + 2n - 2) \]
\[ 288 = \frac{n}{2} \times (2n + 4) \]
\[ 288 = n(n + 2) \]

5. **Rearranging the Equation**:
Rearranging gives us a quadratic equation:
\[ n^2 + 2n - 288 = 0 \]

6. **Using the Quadratic Formula**:
The quadratic formula is:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 2 \), and \( c = -288 \):
\[ n = \frac{-2 \pm \sqrt{2^2 - 4 \times 1 \times (-288)}}{2 \times 1} \]
\[ n = \frac{-2 \pm \sqrt{4 + 1152}}{2} \]
\[ n = \frac{-2 \pm \sqrt{1156}}{2} \]
\[ n = \frac{-2 \pm 34}{2} \]

7. **Calculating the Roots**:
We have two potential solutions:
- \( n = \frac{32}{2} = 16 \)
- \( n = \frac{-36}{2} = -18 \) (not valid since n cannot be negative)

8. **Final Answer**:
Therefore, the number of rows is \( n = 16 \).