Find the sum of G.P. :
`sqrt(3)+(1)/(sqrt(3))+(1)/(3sqrt(3))+ . . . . . . . .` to n terms.

Answer

To find the sum of the given geometric progression (G.P.):

**Given G.P.:**
\[ \sqrt{3} + \frac{1}{\sqrt{3}} + \frac{1}{3\sqrt{3}} + \ldots \]

**Step 1: Identify the first term (a) and the common ratio (r).**

- The first term \( a \) is \( \sqrt{3} \).
- The second term \( a_2 \) is \( \frac{1}{\sqrt{3}} \).

To find the common ratio \( r \):
\[ r = \frac{a_2}{a_1} = \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}} = \frac{1}{3} \]

**Step 2: Use the formula for the sum of the first n terms of a G.P.**

The formula for the sum \( S_n \) of the first \( n \) terms of a G.P. is given by:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]

**Step 3: Substitute the values of \( a \) and \( r \) into the formula.**

Substituting \( a = \sqrt{3} \) and \( r = \frac{1}{3} \):
\[ S_n = \frac{\sqrt{3}(1 - (\frac{1}{3})^n)}{1 - \frac{1}{3}} \]

**Step 4: Simplify the denominator.**

The denominator simplifies as follows:
\[ 1 - \frac{1}{3} = \frac{2}{3} \]

**Step 5: Substitute the denominator back into the equation.**

Now substituting this back into the equation for \( S_n \):
\[ S_n = \frac{\sqrt{3}(1 - (\frac{1}{3})^n)}{\frac{2}{3}} = \sqrt{3}(1 - (\frac{1}{3})^n) \cdot \frac{3}{2} \]

**Step 6: Final simplification.**

This simplifies to:
\[ S_n = \frac{3\sqrt{3}}{2}(1 - (\frac{1}{3})^n) \]

Thus, the sum of the G.P. to \( n \) terms is:
\[ S_n = \frac{3\sqrt{3}}{2} \left( 1 - \left( \frac{1}{3} \right)^n \right) \]

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