Sum up to n terms the series
`(3)/(2)-(5)/(6) +(7)/(18)...`

Answer

To find the sum of the series \( S = \frac{3}{2} - \frac{5}{6} + \frac{7}{18} - \ldots \) up to \( n \) terms, we can follow these steps:

### Step 1: Identify the series
The series is given as:
\[ S = \frac{3}{2} - \frac{5}{6} + \frac{7}{18} - \ldots \]

### Step 2: Multiply the series by \( \frac{1}{3} \)
Multiply the entire series by \( \frac{1}{3} \):
\[ \frac{S}{3} = \frac{3}{6} - \frac{5}{18} + \frac{7}{54} - \ldots \]

### Step 3: Write the equations
Now we have two equations:
1. \( S = \frac{3}{2} - \frac{5}{6} + \frac{7}{18} - \ldots \) (Equation 1)
2. \( \frac{S}{3} = \frac{3}{6} - \frac{5}{18} + \frac{7}{54} - \ldots \) (Equation 2)

### Step 4: Add the two equations
Add Equation 1 and Equation 2:
\[ S + \frac{S}{3} = \left( \frac{3}{2} - \frac{5}{6} + \frac{7}{18} - \ldots \right) + \left( \frac{3}{6} - \frac{5}{18} + \frac{7}{54} - \ldots \right) \]

### Step 5: Simplify the left side
The left side becomes:
\[ \frac{4S}{3} \]

### Step 6: Simplify the right side
On the right side, combine the fractions:
- The first term is \( \frac{3}{2} \).
- The second term becomes \( -\left( \frac{5}{6} - \frac{3}{6} \right) = -\frac{2}{6} = -\frac{1}{3} \).
- The third term becomes \( \left( \frac{7}{18} - \frac{5}{18} \right) = \frac{2}{18} = \frac{1}{9} \).

Thus, the right side simplifies to:
\[ \frac{3}{2} - \frac{1}{3} + \frac{1}{9} - \ldots \]

### Step 7: Recognize the pattern
The series \( -\frac{1}{3} + \frac{1}{9} - \ldots \) can be recognized as a geometric series with first term \( a = -\frac{1}{3} \) and common ratio \( r = -\frac{1}{3} \).

### Step 8: Sum the geometric series
The sum of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
Substituting \( a = -\frac{1}{3} \) and \( r = -\frac{1}{3} \):
\[ S = \frac{-\frac{1}{3}}{1 - (-\frac{1}{3})} = \frac{-\frac{1}{3}}{1 + \frac{1}{3}} = \frac{-\frac{1}{3}}{\frac{4}{3}} = -\frac{1}{4} \]

### Step 9: Substitute back into the equation
Now substitute this back into our equation:
\[ \frac{4S}{3} = \frac{3}{2} + \frac{1}{4} \]
Finding a common denominator (which is 4):
\[ \frac{3}{2} = \frac{6}{4} \]
Thus:
\[ \frac{4S}{3} = \frac{6}{4} - \frac{1}{4} = \frac{5}{4} \]

### Step 10: Solve for \( S \)
Multiply both sides by \( \frac{3}{4} \):
\[ S = \frac{5}{4} \cdot \frac{3}{4} = \frac{15}{16} \]

### Final Result
The sum of the series up to \( n \) terms is:
\[ S = \frac{15}{16} \]

---