How many terms of the G.P.
`(2)/(9),-(1)/(3),(1)/(2), . . . . . . . . . . .` must be added to get the sum equal to `(55)/(72)` ?

Answer

To find the number of terms of the geometric progression (G.P.) \( \frac{2}{9}, -\frac{1}{3}, \frac{1}{2}, \ldots \) that must be added to get a sum equal to \( \frac{55}{72} \), we will follow these steps:

### Step 1: Identify the first term and common ratio
The first term \( a \) of the G.P. is:
\[ a = \frac{2}{9} \]
To find the common ratio \( r \), we divide the second term by the first term:
\[ r = \frac{-\frac{1}{3}}{\frac{2}{9}} = -\frac{1}{3} \times \frac{9}{2} = -\frac{3}{2} \]

### Step 2: Use the formula for the sum of the first \( n \) terms of a G.P.
The formula for the sum of the first \( n \) terms \( S_n \) of a G.P. when \( |r| < 1 \) is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
In this case, since \( r = -\frac{3}{2} \) (which is greater than 1 in absolute value), we will use the formula:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
Setting \( S_n = \frac{55}{72} \), we have:
\[ \frac{55}{72} = \frac{2}{9} \cdot \frac{1 - \left(-\frac{3}{2}\right)^n}{1 - \left(-\frac{3}{2}\right)} \]

### Step 3: Simplify the equation
The denominator simplifies as follows:
\[ 1 - \left(-\frac{3}{2}\right) = 1 + \frac{3}{2} = \frac{2}{2} + \frac{3}{2} = \frac{5}{2} \]
Thus, we can rewrite the equation:
\[ \frac{55}{72} = \frac{2}{9} \cdot \frac{1 - \left(-\frac{3}{2}\right)^n}{\frac{5}{2}} \]
Multiplying both sides by \( \frac{5}{2} \):
\[ \frac{55 \cdot 5}{72 \cdot 2} = \frac{2}{9} \left(1 - \left(-\frac{3}{2}\right)^n\right) \]
Calculating the left side:
\[ \frac{275}{144} = \frac{2}{9} \left(1 - \left(-\frac{3}{2}\right)^n\right) \]

### Step 4: Cross-multiply and simplify
Cross-multiplying gives:
\[ 275 \cdot 9 = 144 \cdot 2 \left(1 - \left(-\frac{3}{2}\right)^n\right) \]
This simplifies to:
\[ 2475 = 288 \left(1 - \left(-\frac{3}{2}\right)^n\right) \]
Dividing both sides by 288:
\[ \frac{2475}{288} = 1 - \left(-\frac{3}{2}\right)^n \]
Calculating \( \frac{2475}{288} \):
\[ 1 - \left(-\frac{3}{2}\right)^n = \frac{2475}{288} \]
Thus:
\[ \left(-\frac{3}{2}\right)^n = 1 - \frac{2475}{288} = \frac{288 - 2475}{288} = \frac{-2187}{288} \]

### Step 5: Solve for \( n \)
We can express \( -2187 \) as \( -3^7 \):
\[ \left(-\frac{3}{2}\right)^n = -\frac{3^7}{288} \]
This implies:
\[ n = 7 \]

### Final Answer
Thus, the required number of terms is:
\[ \boxed{7} \]