If `k ,\ 2k-1` and `2k+1` are three consecutive terms of an A.P., the value of `k` is

Options

A. `-2`
B. `3`
C. `-3`
D. `6`

Answer

To solve the problem, we need to determine the value of \( k \) such that \( k \), \( 2k - 1 \), and \( 2k + 1 \) are three consecutive terms of an Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding the condition for A.P.:**
In an A.P., the difference between consecutive terms is constant. Therefore, the difference between the second term and the first term should be equal to the difference between the third term and the second term.

This can be expressed as:
\[ (2k - 1) - k = (2k + 1) - (2k - 1) \]

2. **Simplifying the left side:**
Calculate the left side:
\[ (2k - 1) - k = 2k - 1 - k = k - 1 \]

3. **Simplifying the right side:**
Calculate the right side:
\[ (2k + 1) - (2k - 1) = 2k + 1 - 2k + 1 = 2 \]

4. **Setting the two sides equal:**
Now we set the two sides equal to each other:
\[ k - 1 = 2 \]

5. **Solving for \( k \):**
To find \( k \), we add 1 to both sides:
\[ k = 2 + 1 = 3 \]

Thus, the value of \( k \) is \( 3 \).

### Final Answer:
The value of \( k \) is \( 3 \).

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