Find the sum of the GP:
(i) `1+3+9+27+...` to 7 terms
(ii) `1+sqrt(3)+3+3sqrt(3)+...` to 10 terms
(iii) `0.15+0.015+0.0015+...` to 6 terms
(iv) `1-1/2+1/4-1/8+...` to 9 terms
(v) `sqrt(2)+1/sqrt(2)+1/(2sqrt(2))+...` to 8 terms
(vi) `2/9-1/3+1/2-3/4+...` to 6 terms

Explanation

To solve the given problem of finding the sum of various geometric progressions (GP), we will use the formula for the sum of the first \( n \) terms of a GP, which is:

\[ S_n = A \frac{1 - R^n}{1 - R} \]

where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( A \) is the first term,
- \( R \) is the common ratio,
- \( n \) is the number of terms.

Let's solve each part step by step.

### Part (i): Find the sum of the GP \( 1 + 3 + 9 + 27 + \ldots \) to 7 terms.

1. Identify \( A \) and \( R \):
- \( A = 1 \)
- \( R = 3 \) (since \( 3/1 = 3 \))

2. Use the formula:
\[ S_7 = 1 \frac{1 - 3^7}{1 - 3} \]

3. Calculate \( 3^7 \):
\[ 3^7 = 2187 \]

4. Substitute back into the formula:
\[ S_7 = 1 \frac{1 - 2187}{1 - 3} = 1 \frac{-2186}{-2} = 1093 \]

**Sum for Part (i): \( S_7 = 1093 \)**

### Part (ii): Find the sum of the GP \( 1 + \sqrt{3} + 3 + 3\sqrt{3} + \ldots \) to 10 terms.

1. Identify \( A \) and \( R \):
- \( A = 1 \)
- \( R = \sqrt{3} \)

2. Use the formula:
\[ S_{10} = 1 \frac{1 - (\sqrt{3})^{10}}{1 - \sqrt{3}} \]

3. Calculate \( (\sqrt{3})^{10} = 3^5 = 243 \):
\[ S_{10} = 1 \frac{1 - 243}{1 - \sqrt{3}} = \frac{-242}{1 - \sqrt{3}} \]

4. Rationalize the denominator:
\[ S_{10} = \frac{-242(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{-242(1 + \sqrt{3})}{3 - 1} = \frac{-121(1 + \sqrt{3})}{1} \]

**Sum for Part (ii): \( S_{10} = -121(1 + \sqrt{3}) \)**

### Part (iii): Find the sum of the GP \( 0.15 + 0.015 + 0.0015 + \ldots \) to 6 terms.

1. Identify \( A \) and \( R \):
- \( A = 0.15 \)
- \( R = 0.1 \)

2. Use the formula:
\[ S_6 = 0.15 \frac{1 - (0.1)^6}{1 - 0.1} \]

3. Calculate \( (0.1)^6 = 0.000001 \):
\[ S_6 = 0.15 \frac{1 - 0.000001}{0.9} = 0.15 \frac{0.999999}{0.9} = 0.15 \times \frac{999999}{900000} = 0.1665 \]

**Sum for Part (iii): \( S_6 = 0.1665 \)**

### Part (iv): Find the sum of the GP \( 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \ldots \) to 9 terms.

1. Identify \( A \) and \( R \):
- \( A = 1 \)
- \( R = -\frac{1}{2} \)

2. Use the formula:
\[ S_9 = 1 \frac{1 - (-\frac{1}{2})^9}{1 - (-\frac{1}{2})} \]

3. Calculate \( (-\frac{1}{2})^9 = -\frac{1}{512} \):
\[ S_9 = 1 \frac{1 + \frac{1}{512}}{1 + \frac{1}{2}} = \frac{1 + \frac{1}{512}}{\frac{3}{2}} = \frac{2 + \frac{1}{256}}{3} = \frac{512 + 1}{768} = \frac{513}{768} \]

**Sum for Part (iv): \( S_9 = \frac{513}{768} \)**

### Part (v): Find the sum of the GP \( \sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + \ldots \) to 8 terms.

1. Identify \( A \) and \( R \):
- \( A = \sqrt{2} \)
- \( R = \frac{1}{\sqrt{2}} \)

2. Use the formula:
\[ S_8 = \sqrt{2} \frac{1 - (\frac{1}{\sqrt{2}})^8}{1 - \frac{1}{\sqrt{2}}} \]

3. Calculate \( (\frac{1}{\sqrt{2}})^8 = \frac{1}{16} \):
\[ S_8 = \sqrt{2} \frac{1 - \frac{1}{16}}{1 - \frac{1}{\sqrt{2}}} = \sqrt{2} \frac{\frac{15}{16}}{1 - \frac{1}{\sqrt{2}}} \]

4. Simplify:
\[ S_8 = \sqrt{2} \frac{15}{16} \cdot \frac{\sqrt{2}}{\sqrt{2} - 1} = \frac{15 \cdot 2}{16(\sqrt{2} - 1)} = \frac{30}{16(\sqrt{2} - 1)} \]

**Sum for Part (v): \( S_8 = \frac{30}{16(\sqrt{2} - 1)} \)**

### Part (vi): Find the sum of the GP \( \frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + \ldots \) to 6 terms.

1. Identify \( A \) and \( R \):
- \( A = \frac{2}{9} \)
- \( R = -\frac{3}{2} \)

2. Use the formula:
\[ S_6 = \frac{2}{9} \frac{1 - (-\frac{3}{2})^6}{1 - (-\frac{3}{2})} \]

3. Calculate \( (-\frac{3}{2})^6 = \frac{729}{64} \):
\[ S_6 = \frac{2}{9} \frac{1 - \frac{729}{64}}{1 + \frac{3}{2}} = \frac{2}{9} \frac{\frac{64 - 729}{64}}{\frac{5}{2}} = \frac{2}{9} \cdot \frac{-665/64}{5/2} = \frac{-133}{144} \]

**Sum for Part (vi): \( S_6 = -\frac{133}{144} \)**

### Summary of Results:
- (i) \( S_7 = 1093 \)
- (ii) \( S_{10} = -121(1 + \sqrt{3}) \)
- (iii) \( S_6 = 0.1665 \)
- (iv) \( S_9 = \frac{513}{768} \)
- (v) \( S_8 = \frac{30}{16(\sqrt{2} - 1)} \)
- (vi) \( S_6 = -\frac{133}{144} \)