For what value of 'K' ; K + 2, 4K – 6, 3K – 2 are three consecutive terms of an A.Ρ.?

Question

Accepted Answer

To determine the value of 'K' for which K + 2, 4K – 6, and 3K – 2 are three consecutive terms of an arithmetic progression (A.P.), we need to use the property of A.P. that the difference between consecutive terms is constant.

### Explanation for Option 3:

1. **Arithmetic Progression Condition:**
   - In an A.P., the difference between consecutive terms is constant. Therefore, for the terms K + 2, 4K – 6, and 3K – 2 to be in A.P., the difference between the first and second terms should be equal to the difference between the second and third terms.

2. **Set up the Equation:**
   - The difference between the second term and the first term is: 
     $$(4K - 6) - (K + 2) = 4K - 6 - K - 2 = 3K - 8$$
   - The difference between the third term and the second term is:
     $$(3K - 2) - (4K - 6) = 3K - 2 - 4K + 6 = -K + 4$$

3. **Equate the Differences:**
   - Since these differences must be equal for the terms to be in A.P., we set them equal to each other:
     $$3K - 8 = -K + 4$$

4. **Solve for K:**
   - Add K to both sides:
     $$3K + K - 8 = 4$$
     $$4K - 8 = 4$$
   - Add 8 to both sides:
     $$4K = 12$$
   - Divide by 4:
     $$K = 3$$

Thus, the correct value of K is 3, which corresponds to Option 3.

### Explanation for Why Other Options are Incorrect:

1. **Option 1 (K = 1):**
   - If K = 1, the terms become 3, -2, and 1.
   - The differences are: 
     $$-2 - 3 = -5$$ and $$1 - (-2) = 3$$
   - Since -5 ≠ 3, these terms are not in A.P.

2. **Option 2 (K = -1):**
   - If K = -1, the terms become 1, -10, and -5.
   - The differences are:
     $$-10 - 1 = -11$$ and $$-5 - (-10) = 5$$
   - Since -11 ≠ 5, these terms are not in A.P.

3. **Option 4 (K = -3):**
   - If K = -3, the terms become -1, -18, and -11.
   - The differences are:
     $$-18 - (-1) = -17$$ and $$-11 - (-18) = 7$$
   - Since -17 ≠ 7, these terms are not in A.P.

Therefore, Option 3 is the only correct choice, as it satisfies the condition for the terms to be in an arithmetic progression.