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

Options

A. `6`
B. `3`
C. `4`
D. `5`

Explanation

To solve the problem of how many terms of the geometric progression (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 common ratio
The first term \( a \) of the G.P. is:
\[ a = \frac{2}{9} \]

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

### Step 2: Use the formula for the sum of the first \( n \) terms of a G.P.
The formula for the sum \( S_n \) of the first \( n \) terms of a G.P. when \( |r| < 1 \) is given by:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
In this case, we have:
\[ S_n = \frac{\frac{2}{9}(1 - (-\frac{3}{2})^n)}{1 - (-\frac{3}{2})} \]

### Step 3: Simplify the denominator
Calculating the denominator:
\[ 1 - (-\frac{3}{2}) = 1 + \frac{3}{2} = \frac{2}{2} + \frac{3}{2} = \frac{5}{2} \]

### Step 4: Substitute into the sum formula
Now substituting back into the sum formula:
\[ S_n = \frac{\frac{2}{9}(1 - (-\frac{3}{2})^n)}{\frac{5}{2}} = \frac{2}{9} \cdot \frac{2}{5} (1 - (-\frac{3}{2})^n) = \frac{4}{45}(1 - (-\frac{3}{2})^n) \]

### Step 5: Set the sum equal to \( \frac{55}{72} \)
We set the sum equal to the given value:
\[ \frac{4}{45}(1 - (-\frac{3}{2})^n) = \frac{55}{72} \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 4 \cdot 72 (1 - (-\frac{3}{2})^n) = 55 \cdot 45 \]
Calculating both sides:
\[ 288(1 - (-\frac{3}{2})^n) = 2475 \]

### Step 7: Solve for \( (-\frac{3}{2})^n \)
Dividing both sides by 288:
\[ 1 - (-\frac{3}{2})^n = \frac{2475}{288} \]
This simplifies to:
\[ (-\frac{3}{2})^n = 1 - \frac{2475}{288} = \frac{288 - 2475}{288} = \frac{-2187}{288} \]

### Step 8: Express \( -2187 \) as a power of \( -\frac{3}{2} \)
We know that:
\[ -2187 = -\left(\frac{3}{2}\right)^5 \]
Thus:
\[ (-\frac{3}{2})^n = -\left(\frac{3}{2}\right)^5 \]

### Step 9: Equate the powers
Since the bases are the same, we equate the exponents:
\[ n = 5 \]

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