What does the series `1+3^(-1//2)+3+(1)/(3sqrt3)+` . . .represent ?

Options

A. AP
B. GP
C. HP
D. None of these

Answer

To determine what the series \(1 + 3^{-1/2} + 3 + \frac{1}{3\sqrt{3}} + \ldots\) represents, we will analyze the terms step by step.

### Step 1: Rewrite the Series
First, let's rewrite the series in a more recognizable form:
\[ 1 + \frac{1}{\sqrt{3}} + 3 + \frac{1}{3\sqrt{3}} + \ldots \]

### Step 2: Identify the Terms
Now, we can identify the terms:
- First term \(a_1 = 1\)
- Second term \(a_2 = \frac{1}{\sqrt{3}}\)
- Third term \(a_3 = 3\)
- Fourth term \(a_4 = \frac{1}{3\sqrt{3}}\)

### Step 3: Check for Arithmetic Progression (AP)
For a series to be in Arithmetic Progression (AP), the difference between consecutive terms must be constant. Let's calculate the differences:
- \(a_2 - a_1 = \frac{1}{\sqrt{3}} - 1\)
- \(a_3 - a_2 = 3 - \frac{1}{\sqrt{3}}\)
- \(a_4 - a_3 = \frac{1}{3\sqrt{3}} - 3\)

Since these differences are not constant, the series is not in AP.

### Step 4: Check for Geometric Progression (GP)
For a series to be in Geometric Progression (GP), the ratio of consecutive terms must be constant. Let's calculate the ratios:
- \(r_1 = \frac{a_2}{a_1} = \frac{\frac{1}{\sqrt{3}}}{1} = \frac{1}{\sqrt{3}}\)
- \(r_2 = \frac{a_3}{a_2} = \frac{3}{\frac{1}{\sqrt{3}}} = 3\sqrt{3}\)
- \(r_3 = \frac{a_4}{a_3} = \frac{\frac{1}{3\sqrt{3}}}{3} = \frac{1}{9\sqrt{3}}\)

Since these ratios are not constant, the series is not in GP.

### Step 5: Check for Harmonic Progression (HP)
For a series to be in Harmonic Progression (HP), the reciprocals of the terms must be in AP. Let's find the reciprocals:
- \(b_1 = 1\)
- \(b_2 = \sqrt{3}\)
- \(b_3 = \frac{1}{3}\)
- \(b_4 = 3\sqrt{3}\)

Now, we check if \(b_1, b_2, b_3, b_4\) are in AP:
- The differences are not constant, hence the series is not in HP.

### Conclusion
Since the series does not fit into AP, GP, or HP, we conclude that it does not represent any of these progressions.

Thus, the answer is that the series represents none of the given options.