How many terms of the series `(2)/(3)-1+(3)/(2)`… has the sum `(463)/(96)`?

Explanation

To solve the problem of finding how many terms of the series \( \frac{2}{3}, -1, \frac{3}{2}, \ldots \) sum up to \( \frac{463}{96} \), we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \( a \) of the series is:
\[ a = \frac{2}{3} \]

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

### 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:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]

### Step 3: Substitute the known values into the formula
Substituting \( a = \frac{2}{3} \) and \( r = -\frac{3}{2} \) into the formula:
\[ S_n = \frac{\frac{2}{3}(1 - (-\frac{3}{2})^n)}{1 - (-\frac{3}{2})} \]

### Step 4: Simplify the denominator
Calculating the denominator:
\[ 1 - (-\frac{3}{2}) = 1 + \frac{3}{2} = \frac{2}{2} + \frac{3}{2} = \frac{5}{2} \]

Now substituting this back into the formula:
\[ S_n = \frac{\frac{2}{3}(1 - (-\frac{3}{2})^n)}{\frac{5}{2}} = \frac{2}{3} \cdot \frac{2}{5} (1 - (-\frac{3}{2})^n) = \frac{4}{15}(1 - (-\frac{3}{2})^n) \]

### Step 5: Set the sum equal to \( \frac{463}{96} \)
We know that:
\[ S_n = \frac{463}{96} \]
So we set up the equation:
\[ \frac{4}{15}(1 - (-\frac{3}{2})^n) = \frac{463}{96} \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 4 \cdot 96 (1 - (-\frac{3}{2})^n) = 463 \cdot 15 \]
Calculating the left side:
\[ 384(1 - (-\frac{3}{2})^n) = 6945 \]

### Step 7: Isolate the term involving \( n \)
Dividing both sides by 384:
\[ 1 - (-\frac{3}{2})^n = \frac{6945}{384} \]
Calculating \( \frac{6945}{384} \) gives:
\[ 1 - (-\frac{3}{2})^n = \frac{2315}{128} \]

### Step 8: Solve for \( (-\frac{3}{2})^n \)
Rearranging gives:
\[ (-\frac{3}{2})^n = 1 - \frac{2315}{128} = \frac{128 - 2315}{128} = \frac{-2187}{128} \]

### Step 9: Express the left side as powers
Since \( (-\frac{3}{2})^n = -\frac{3^n}{2^n} \), we have:
\[ -\frac{3^n}{2^n} = \frac{-2187}{128} \]
This implies:
\[ \frac{3^n}{2^n} = \frac{2187}{128} \]

### Step 10: Recognize the powers of 3 and 2
Notice that \( 2187 = 3^7 \) and \( 128 = 2^7 \), thus:
\[ \frac{3^n}{2^n} = \frac{3^7}{2^7} \]

### Step 11: Set the exponents equal
This gives us:
\[ n = 7 \]

### Final Answer
Thus, the number of terms of the series that sum to \( \frac{463}{96} \) is:
\[ \boxed{7} \]