Of how many terms is ,`(55)/(72)` the sum of the series `(2)/(9) -(1)/(3)+(1)/(2)- ....` ?

Answer

To solve the problem of finding how many terms of the series \( \frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \ldots \) sum up to \( \frac{55}{72} \), we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \( a \) of the series is:
\[ a = \frac{2}{9} \]

To find the common ratio \( r \), we can take the second term and divide it by the first term:
\[ r = \frac{-\frac{1}{3}}{\frac{2}{9}} = -\frac{1}{3} \times \frac{9}{2} = -\frac{9}{6} = -\frac{3}{2} \]

### Step 2: Write the formula for the sum of the first \( n \) terms of a geometric series
The formula for the sum of the first \( n \) terms \( S_n \) of a geometric series is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substituting the values of \( a \) and \( r \):
\[ S_n = \frac{2}{9} \frac{(-\frac{3}{2})^n - 1}{-\frac{3}{2} - 1} \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:
\[ -\frac{3}{2} - 1 = -\frac{3}{2} - \frac{2}{2} = -\frac{5}{2} \]
Thus, we can rewrite \( S_n \):
\[ S_n = \frac{2}{9} \frac{(-\frac{3}{2})^n - 1}{-\frac{5}{2}} = \frac{2}{9} \cdot \frac{2}{-5} \cdot \left((-3/2)^n - 1\right) \]
This simplifies to:
\[ S_n = \frac{4}{-45} \left((-3/2)^n - 1\right) = -\frac{4}{45} \left((-3/2)^n - 1\right) \]

### Step 4: Set the sum equal to \( \frac{55}{72} \)
We need to find \( n \) such that:
\[ -\frac{4}{45} \left((-3/2)^n - 1\right) = \frac{55}{72} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ -4 \left((-3/2)^n - 1\right) \cdot 72 = 55 \cdot 45 \]
This simplifies to:
\[ -288 \left((-3/2)^n - 1\right) = 2475 \]

### Step 6: Solve for \( (-3/2)^n \)
Dividing both sides by -288:
\[ (-3/2)^n - 1 = -\frac{2475}{288} \]
Thus:
\[ (-3/2)^n = 1 - \frac{2475}{288} \]
Calculating the right side:
\[ 1 = \frac{288}{288} \implies (-3/2)^n = \frac{288 - 2475}{288} = \frac{-2187}{288} \]

### Step 7: Recognize powers of 3 and 2
We can express \( -2187 \) as \( -3^7 \) and \( 288 \) as \( 2^5 \cdot 3^2 \):
\[ (-3/2)^n = \frac{-3^7}{2^5 \cdot 3^2} \]
This implies:
\[ (-3/2)^n = -\left(\frac{3}{2}\right)^n \]
Thus:
\[ n = 7 \]

### Conclusion
The number of terms for which the sum is \( \frac{55}{72} \) is:
\[ \boxed{7} \]