Which one is an A.P. series ?

Options

A. `a, a^2, a^3,….`
B. `1^2, 3^2 , 5^2, 7^2 ,….`
C. `a, 2a, 3a, 4a, …….`
D. `1,3,9,27`,…..

Answer

To determine which of the given options is an Arithmetic Progression (A.P.), we need to check if the difference between consecutive terms is constant for each option. An A.P. is defined by having a common difference \( d \) such that:

\[ d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = \ldots \]

Let's analyze each option step by step.

### Step 1: Analyze the First Option
**First Option:** \( a, a^2, a^3 \)

- Calculate the differences:
- \( a^2 - a = a(a - 1) \)
- \( a^3 - a^2 = a^2(a - 1) \)

- Since \( a(a - 1) \) and \( a^2(a - 1) \) are not equal for all \( a \), this option is **not an A.P.**

### Step 2: Analyze the Second Option
**Second Option:** \( 1^2, 3^2, 5^2, 7^2 \)

- Calculate the differences:
- \( 3^2 - 1^2 = 9 - 1 = 8 \)
- \( 5^2 - 3^2 = 25 - 9 = 16 \)
- \( 7^2 - 5^2 = 49 - 25 = 24 \)

- The differences are \( 8, 16, 24 \), which are not equal. Thus, this option is also **not an A.P.**

### Step 3: Analyze the Third Option
**Third Option:** \( a, 2a, 3a, 4a \)

- Calculate the differences:
- \( 2a - a = a \)
- \( 3a - 2a = a \)
- \( 4a - 3a = a \)

- All differences are equal to \( a \). Therefore, this option **is an A.P.**

### Step 4: Analyze the Fourth Option
**Fourth Option:** \( 1, 3, 9, 27 \)

- Calculate the differences:
- \( 3 - 1 = 2 \)
- \( 9 - 3 = 6 \)
- \( 27 - 9 = 18 \)

- The differences are \( 2, 6, 18 \), which are not equal. Thus, this option is also **not an A.P.**

### Conclusion
The only option that forms an Arithmetic Progression is the **Third Option: \( a, 2a, 3a, 4a \)**.

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