Consider the pattern shown below:
`{:(" Row ",1,1,,,),(" Row ",2,3,5,,),(" Row ",3,7,9,11,),(" Row ",4,13,15,17,19):}etc.`
The number at the end of row 60 is

Options

A. 3659
B. 3519
C. 3681
D. 3731

Explanation

To find the number at the end of row 60 in the given pattern, we will derive a formula for the last number in the nth row, denoted as T(n).

### Step-by-Step Solution:

1. **Identify the Last Numbers in Each Row**:
- Row 1: 1
- Row 2: 5
- Row 3: 11
- Row 4: 19

2. **Find the Differences**:
- The differences between consecutive last numbers:
- 5 - 1 = 4
- 11 - 5 = 6
- 19 - 11 = 8
- The differences are: 4, 6, 8, which form an arithmetic progression (AP) with a common difference of 2.

3. **Express the Last Number in Terms of n**:
- Let T(n) be the last number in the nth row.
- We can express T(n) as:
\[ T(n) = T(n-1) + (4 + 2(n-2)) \]
- This means that each term is derived from the previous term plus an increasing amount.

4. **Establish a Formula**:
- We can derive a formula for T(n) based on the pattern observed:
- From the differences, we can see that:
\[ T(n) = T(1) + \text{sum of differences up to } n-1 \]
- The sum of the first (n-1) terms of the AP (4, 6, 8, ...) can be calculated as:
\[ \text{Sum} = \frac{(n-1)}{2} \times (2 \times 4 + (n-2) \times 2) \]
- Simplifying gives:
\[ T(n) = 1 + \frac{(n-1)}{2} \times (8 + 2(n-2)) \]
- This simplifies to:
\[ T(n) = 1 + (n-1)(n + 1) \]
- Finally, we arrive at:
\[ T(n) = n^2 + n - 1 \]

5. **Calculate T(60)**:
- Substitute n = 60 into the formula:
\[ T(60) = 60^2 + 60 - 1 \]
- Calculate:
\[ T(60) = 3600 + 60 - 1 = 3659 \]

### Final Answer:
The number at the end of row 60 is **3659**.