How many terms of the G.P. `(2)/(9)-(1)/(3)+(1)/(2)..." give the sum "(55)/(72)?`

Explanation

To solve the problem of how many terms of the G.P. \( \frac{2}{9}, -\frac{1}{3}, \frac{1}{2}, \ldots \) give the sum \( \frac{55}{72} \), we will follow these steps:

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

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

### Step 2: Write 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:
\[ S_n = \frac{A(R^n - 1)}{R - 1} \]
Substituting the values of \( A \) and \( R \):
\[ S_n = \frac{\frac{2}{9} \left( \left(-\frac{3}{2}\right)^n - 1 \right)}{-\frac{3}{2} - 1} \]

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

### Step 4: Set the sum equal to \( \frac{55}{72} \)
We set the expression for \( S_n \) equal to \( \frac{55}{72} \):
\[ \frac{-4}{45} \left( \left(-\frac{3}{2}\right)^n - 1 \right) = \frac{55}{72} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ -4 \left( \left(-\frac{3}{2}\right)^n - 1 \right) \cdot 72 = 55 \cdot 45 \]
Calculating the right side:
\[ 55 \cdot 45 = 2475 \]
Thus, we have:
\[ -288 \left( \left(-\frac{3}{2}\right)^n - 1 \right) = 2475 \]

### Step 6: Isolate the term with \( n \)
Dividing both sides by -288:
\[ \left(-\frac{3}{2}\right)^n - 1 = -\frac{2475}{288} \]
Adding 1 to both sides:
\[ \left(-\frac{3}{2}\right)^n = 1 - \frac{2475}{288} \]
Calculating the right side:
\[ 1 = \frac{288}{288} \quad \Rightarrow \quad \left(-\frac{3}{2}\right)^n = \frac{288 - 2475}{288} = \frac{-2187}{288} \]

### Step 7: Solve for \( n \)
Recognizing that \( -2187 = -3^7 \) and \( 288 = 2^5 \cdot 3^2 \):
\[ \left(-\frac{3}{2}\right)^n = -\frac{3^7}{2^5 \cdot 3^2} \]
This simplifies to:
\[ \left(-\frac{3}{2}\right)^n = -\frac{3^{7-2}}{2^5} = -\frac{3^5}{2^5} \]
Thus, we have:
\[ n = 5 \]

### Final Answer
The number of terms of the G.P. that give the sum \( \frac{55}{72} \) is \( n = 5 \).