How many terms are needed of the series `(2)/(9)-(1)/(3)+(1)/(2)-…` to give the sum `(55)/(72)`?

Options

A. 8
B. 7
C. 5
D. 4

Explanation

To find the number of terms needed in the series \( \frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \ldots \) to achieve a sum of \( \frac{55}{72} \), we will 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:
\[ r = \frac{-\frac{1}{3}}{\frac{2}{9}} = -\frac{1}{3} \cdot \frac{9}{2} = -\frac{3}{2} \]

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

### 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 the denominator back 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 \( \frac{55}{72} \):
\[ \frac{4}{45}(1 - (-\frac{3}{2})^n) = \frac{55}{72} \]

### 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: Isolate the term with \( n \)
Dividing both sides by 288:
\[ 1 - (-\frac{3}{2})^n = \frac{2475}{288} \]
Calculating \( \frac{2475}{288} \):
\[ 1 - (-\frac{3}{2})^n = \frac{2475}{288} \implies (-\frac{3}{2})^n = 1 - \frac{2475}{288} \]
Finding a common denominator:
\[ 1 = \frac{288}{288} \implies (-\frac{3}{2})^n = \frac{288 - 2475}{288} = \frac{-2187}{288} \]

### Step 8: Solve for \( n \)
Since \( (-\frac{3}{2})^n = -\left(\frac{3}{2}\right)^n \):
\[ -\left(\frac{3}{2}\right)^n = \frac{-2187}{288} \]
This implies:
\[ \left(\frac{3}{2}\right)^n = \frac{2187}{288} \]
Now, we can express \( 2187 \) as \( 3^7 \) and \( 288 \) as \( 2^5 \cdot 3^2 \):
\[ \left(\frac{3}{2}\right)^n = \frac{3^7}{2^5 \cdot 3^2} \implies \left(\frac{3}{2}\right)^n = \frac{3^{7-2}}{2^5} = \frac{3^5}{2^5} \]
Thus:
\[ n = 5 \]

### Final Answer
The number of terms needed is \( n = 5 \).

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