Which term of the sequence 6, 13,20,27…….is 98 more than its 24th term

Options

A. 36th
B. 38th
C. 35th
D. 48th

Answer

To solve the problem, we need to find which term of the sequence \(6, 13, 20, 27, \ldots\) is 98 more than its 24th term.

### Step-by-Step Solution:

1. **Identify the Sequence:**
The sequence given is \(6, 13, 20, 27, \ldots\). We can see that the common difference \(d\) is \(7\) (since \(13 - 6 = 7\), \(20 - 13 = 7\), etc.).

2. **Find the 24th Term:**
The formula for the \(n\)th term of an arithmetic sequence is given by:
\[ a_n = a + (n - 1) \cdot d \]
where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

Here, \(a = 6\), \(d = 7\), and \(n = 24\).
\[ a_{24} = 6 + (24 - 1) \cdot 7 \]
\[ = 6 + 23 \cdot 7 \]
\[ = 6 + 161 \]
\[ = 167 \]

3. **Calculate the Value 98 More than the 24th Term:**
We need to find the term that is 98 more than the 24th term:
\[ 167 + 98 = 265 \]

4. **Set Up the Equation for the \(n\)th Term:**
Now we need to find \(n\) such that:
\[ a_n = 265 \]
Using the formula for the \(n\)th term:
\[ 265 = 6 + (n - 1) \cdot 7 \]

5. **Solve for \(n\):**
Rearranging the equation:
\[ 265 - 6 = (n - 1) \cdot 7 \]
\[ 259 = (n - 1) \cdot 7 \]
Dividing both sides by \(7\):
\[ n - 1 = \frac{259}{7} = 37 \]
\[ n = 37 + 1 = 38 \]

6. **Conclusion:**
The term of the sequence that is 98 more than the 24th term is the **38th term**.