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 find the value of \( k \) such that \( k \), \( 2k - 1 \), and \( 2k + 1 \) are three consecutive terms of an arithmetic progression (A.P.), we can follow these steps:

### Step 1: Understand the property of A.P.
In an arithmetic progression, the difference between consecutive terms is constant. Therefore, we can set up the equation based on the property of A.P.:
\[ a_2 - a_1 = a_3 - a_2 \]
where \( a_1 = k \), \( a_2 = 2k - 1 \), and \( a_3 = 2k + 1 \).

### Step 2: Set up the equation
Substituting the values of \( a_1 \), \( a_2 \), and \( a_3 \) into the equation gives:
\[ (2k - 1) - k = (2k + 1) - (2k - 1) \]

### Step 3: Simplify the left side
Now simplify the left side:
\[ 2k - 1 - k = k - 1 \]

### Step 4: Simplify the right side
Now simplify the right side:
\[ (2k + 1) - (2k - 1) = 2 \]

### Step 5: Set the two sides equal
Now we have:
\[ k - 1 = 2 \]

### Step 6: Solve for \( k \)
To find \( k \), add 1 to both sides:
\[ k = 2 + 1 = 3 \]

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

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