If `t_n` denotes the nth term of the series 2+3+6+11+18+….. Then `t_50` is

Options

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

Explanation

To find the 50th term of the series given by \( t_n = 2, 3, 6, 11, 18, \ldots \), we need to first identify the pattern in the series.

### Step 1: Identify the pattern in the series
Let's write down the terms of the series:
- \( t_1 = 2 \)
- \( t_2 = 3 \)
- \( t_3 = 6 \)
- \( t_4 = 11 \)
- \( t_5 = 18 \)

### Step 2: Find the differences between consecutive terms
Now, we can find the differences between consecutive terms:
- \( t_2 - t_1 = 3 - 2 = 1 \)
- \( t_3 - t_2 = 6 - 3 = 3 \)
- \( t_4 - t_3 = 11 - 6 = 5 \)
- \( t_5 - t_4 = 18 - 11 = 7 \)

This gives us the first differences: \( 1, 3, 5, 7 \).

### Step 3: Find the second differences
Next, we calculate the second differences:
- \( 3 - 1 = 2 \)
- \( 5 - 3 = 2 \)
- \( 7 - 5 = 2 \)

The second differences are constant and equal to \( 2 \). This suggests that the sequence is quadratic.

### Step 4: Formulate the general term
Since the second differences are constant, we can express the \( n \)-th term of the series in the form:
\[ t_n = an^2 + bn + c \]

### Step 5: Set up equations using known terms
We can use the first three terms to set up equations:
1. For \( n = 1 \): \( a(1)^2 + b(1) + c = 2 \) → \( a + b + c = 2 \) (Equation 1)
2. For \( n = 2 \): \( a(2)^2 + b(2) + c = 3 \) → \( 4a + 2b + c = 3 \) (Equation 2)
3. For \( n = 3 \): \( a(3)^2 + b(3) + c = 6 \) → \( 9a + 3b + c = 6 \) (Equation 3)

### Step 6: Solve the system of equations
Now we solve these equations step by step.

From Equation 1:
\[ c = 2 - a - b \]

Substituting \( c \) into Equation 2:
\[ 4a + 2b + (2 - a - b) = 3 \]
\[ 3a + b + 2 = 3 \implies 3a + b = 1 \quad (Equation 4) \]

Now substitute \( c \) into Equation 3:
\[ 9a + 3b + (2 - a - b) = 6 \]
\[ 8a + 2b + 2 = 6 \implies 8a + 2b = 4 \implies 4a + b = 2 \quad (Equation 5) \]

### Step 7: Solve Equations 4 and 5
Now we have a system of two equations:
1. \( 3a + b = 1 \) (Equation 4)
2. \( 4a + b = 2 \) (Equation 5)

Subtract Equation 4 from Equation 5:
\[ (4a + b) - (3a + b) = 2 - 1 \]
\[ a = 1 \]

Substituting \( a = 1 \) back into Equation 4:
\[ 3(1) + b = 1 \implies b = -2 \]

Now substitute \( a \) and \( b \) back into Equation 1 to find \( c \):
\[ 1 - 2 + c = 2 \implies c = 3 \]

### Step 8: Write the general term
Thus, the general term is:
\[ t_n = n^2 - 2n + 3 \]

### Step 9: Find \( t_{50} \)
Now we can find \( t_{50} \):
\[ t_{50} = 50^2 - 2(50) + 3 \]
\[ t_{50} = 2500 - 100 + 3 = 2403 \]

### Final Answer
Therefore, \( t_{50} = 2403 \).

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