If `t_n` denotes the nth term of the series `2+3+6+11+18+...`then `t_(50)`=...... `49^2-1` b. `49^2` c. `50^2+1` d. `49^2+2`

Options

A. `49^(2)-1`
B. `49^(2)`
C. `50^(2)+1`
D. `49^(2)+2`

Explanation

To find the 50th term \( t_{50} \) of the series \( 2, 3, 6, 11, 18, \ldots \), we will first identify a pattern in the series.

### Step 1: Write down the given series
The series given is:
\[ 2, 3, 6, 11, 18, \ldots \]

### Step 2: Analyze the terms
We can express each term in a different way to identify a pattern:
- The first term \( t_1 = 2 \) can be rewritten as \( 0^2 + 2 \)
- The second term \( t_2 = 3 \) can be rewritten as \( 1^2 + 2 \)
- The third term \( t_3 = 6 \) can be rewritten as \( 2^2 + 2 \)
- The fourth term \( t_4 = 11 \) can be rewritten as \( 3^2 + 2 \)
- The fifth term \( t_5 = 18 \) can be rewritten as \( 4^2 + 2 \)

### Step 3: Generalize the nth term
From the above observations, we can see a pattern emerging:
\[ t_n = (n-1)^2 + 2 \]
This means that the nth term can be expressed as the square of \( (n-1) \) plus 2.

### Step 4: Find the 50th term
To find \( t_{50} \), we substitute \( n = 50 \) into the formula:
\[ t_{50} = (50 - 1)^2 + 2 \]
\[ t_{50} = 49^2 + 2 \]

### Step 5: Calculate \( 49^2 + 2 \)
Now we calculate \( 49^2 \):
\[ 49^2 = 2401 \]
Thus,
\[ t_{50} = 2401 + 2 = 2403 \]

### Conclusion
The value of \( t_{50} \) is \( 49^2 + 2 \).

### Final Answer
Thus, the answer is:
\[ \boxed{49^2 + 2} \]