The sum of the n terms of a series in `nl + n^2` then the 6th terms is, if `n in N`.

Options

A. a. 756
B. b. 611
C. d. data insufficient
D. d. none of these

Answer

To find the 6th term of the series where the sum of the first n terms is given by \( S_n = nl + n^2 \), we can follow these steps:

### Step 1: Write down the formula for the sum of the first n terms.
The sum of the first n terms is given by:
\[ S_n = nl + n^2 \]

### Step 2: Calculate the sum of the first 6 terms.
Substituting \( n = 6 \) into the formula, we get:
\[ S_6 = 6l + 6^2 \]
\[ S_6 = 6l + 36 \]

### Step 3: Calculate the sum of the first 5 terms.
Now, substituting \( n = 5 \) into the formula, we get:
\[ S_5 = 5l + 5^2 \]
\[ S_5 = 5l + 25 \]

### Step 4: Find the 6th term.
The 6th term \( T_6 \) can be found using the formula:
\[ T_6 = S_6 - S_5 \]
Substituting the values we calculated:
\[ T_6 = (6l + 36) - (5l + 25) \]

### Step 5: Simplify the expression.
Now, simplify the expression:
\[ T_6 = 6l + 36 - 5l - 25 \]
\[ T_6 = (6l - 5l) + (36 - 25) \]
\[ T_6 = l + 11 \]

### Final Answer:
Thus, the 6th term of the series is:
\[ T_6 = l + 11 \]

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