What is the `15^(th)` term of the series 3, 7, 13, 21, 31, 43, …….?

Options

A. 205
B. 225
C. 238
D. 241

Explanation

To find the 15th term of the series 3, 7, 13, 21, 31, 43, ..., we first need to identify the pattern in the series.

### Step 1: Identify the series
The given series is:
- 3, 7, 13, 21, 31, 43, ...

### Step 2: Find the differences between consecutive terms
Let's find the differences between consecutive terms:
- 7 - 3 = 4
- 13 - 7 = 6
- 21 - 13 = 8
- 31 - 21 = 10
- 43 - 31 = 12

The differences are:
- 4, 6, 8, 10, 12, ...

### Step 3: Identify the pattern in the differences
The differences themselves form an arithmetic series:
- The first difference is 4, and the common difference is 2.

### Step 4: General term of the series
We can express the nth term of the series using the formula for the sum of the first n terms of an arithmetic series:
- The nth term can be expressed as:
\[ a_n = a_1 + S_{n-1} \]
where \( S_{n-1} \) is the sum of the first \( n-1 \) differences.

### Step 5: Calculate the sum of the first \( n-1 \) differences
The sum of the first \( n-1 \) terms of the differences can be calculated as:
\[ S_{n-1} = \frac{(n-1)}{2} \times (2a + (n-2)d) \]
where \( a = 4 \) (the first difference) and \( d = 2 \) (the common difference).

### Step 6: Substitute values for \( n = 15 \)
Now we need to find \( a_{15} \):
- The first term \( a_1 = 3 \)
- The number of terms \( n = 15 \)

Calculating \( S_{14} \):
\[ S_{14} = \frac{14}{2} \times (2 \times 4 + (14 - 1) \times 2) \]
\[ = 7 \times (8 + 26) = 7 \times 34 = 238 \]

### Step 7: Find the 15th term
Now we can find \( a_{15} \):
\[ a_{15} = 3 + S_{14} = 3 + 238 = 241 \]

### Final Answer
Thus, the 15th term of the series is:
\[ \boxed{241} \]