50th term of the sequence `3+12+25+42+` is `5145` b. `5148` c. `5142` d. `5195`

Options

A. 5145
B. 5148
C. 5142
D. 5195

Answer

To find the 50th term of the sequence given by the terms 3, 12, 25, 42, we can follow these steps:

### Step 1: Identify the pattern in the sequence
The first four terms are:
- \( T_1 = 3 \)
- \( T_2 = 12 \)
- \( T_3 = 25 \)
- \( T_4 = 42 \)

Next, we calculate the differences between consecutive terms:
- \( T_2 - T_1 = 12 - 3 = 9 \)
- \( T_3 - T_2 = 25 - 12 = 13 \)
- \( T_4 - T_3 = 42 - 25 = 17 \)

The differences are 9, 13, and 17. We can see that the differences themselves are increasing by 4:
- \( 13 - 9 = 4 \)
- \( 17 - 13 = 4 \)

### Step 2: Formulate a general term
Since the second differences are constant, we can assume that the \( n \)-th term of the sequence can be expressed in the form:
\[ T_n = an^2 + bn + c \]

### Step 3: Set up equations
We can set up equations using the first three terms:
1. For \( n = 1 \):
\[ a(1^2) + b(1) + c = 3 \]
\[ a + b + c = 3 \] (Equation 1)

2. For \( n = 2 \):
\[ a(2^2) + b(2) + c = 12 \]
\[ 4a + 2b + c = 12 \] (Equation 2)

3. For \( n = 3 \):
\[ a(3^2) + b(3) + c = 25 \]
\[ 9a + 3b + c = 25 \] (Equation 3)

### Step 4: Solve the equations
Now, we will solve these equations step by step.

**Subtract Equation 1 from Equation 2:**
\[ (4a + 2b + c) - (a + b + c) = 12 - 3 \]
\[ 3a + b = 9 \] (Equation 4)

**Subtract Equation 2 from Equation 3:**
\[ (9a + 3b + c) - (4a + 2b + c) = 25 - 12 \]
\[ 5a + b = 13 \] (Equation 5)

**Now subtract Equation 4 from Equation 5:**
\[ (5a + b) - (3a + b) = 13 - 9 \]
\[ 2a = 4 \]
\[ a = 2 \]

**Substituting \( a = 2 \) back into Equation 4:**
\[ 3(2) + b = 9 \]
\[ 6 + b = 9 \]
\[ b = 3 \]

**Substituting \( a = 2 \) and \( b = 3 \) back into Equation 1:**
\[ 2 + 3 + c = 3 \]
\[ c = 3 - 5 \]
\[ c = -2 \]

### Step 5: Write the general term
Now we have:
\[ T_n = 2n^2 + 3n - 2 \]

### Step 6: Find the 50th term
Now, we can find the 50th term:
\[ T_{50} = 2(50^2) + 3(50) - 2 \]
\[ = 2(2500) + 150 - 2 \]
\[ = 5000 + 150 - 2 \]
\[ = 5148 \]

### Conclusion
Thus, the 50th term of the sequence is **5148**.