The sum of first 50 term of the series ` 1 + (3 ) /(2) + (7) /(4) + (15 ) /(8) + (31 ) /(16) + `….is `( p + (1)/(2^q )) ` , then value of ` ( p + q ) ` is ……

Explanation

To solve the problem, we need to find the sum of the first 50 terms of the series:

\[ S = 1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \ldots \]

### Step 1: Identify the pattern in the series

The numerators of the series are:
- \(1, 3, 7, 15, 31, \ldots\)

We can observe that:
- \(1 = 2^1 - 1\)
- \(3 = 2^2 - 1\)
- \(7 = 2^3 - 1\)
- \(15 = 2^4 - 1\)
- \(31 = 2^5 - 1\)

Thus, the \(n\)-th term in the numerator can be expressed as:
\[ a_n = 2^n - 1 \]

The denominators are:
- \(1, 2, 4, 8, 16, \ldots\)

These can be expressed as:
\[ b_n = 2^{n-1} \]

### Step 2: Write the general term of the series

The \(n\)-th term of the series can be written as:
\[ T_n = \frac{2^n - 1}{2^{n-1}} = 2 - \frac{1}{2^{n-1}} \]

### Step 3: Find the sum of the first 50 terms

The sum of the first 50 terms can be expressed as:
\[ S_{50} = \sum_{n=1}^{50} T_n = \sum_{n=1}^{50} \left( 2 - \frac{1}{2^{n-1}} \right) \]

This can be separated into two sums:
\[ S_{50} = \sum_{n=1}^{50} 2 - \sum_{n=1}^{50} \frac{1}{2^{n-1}} \]

### Step 4: Calculate the first sum

The first sum is simply:
\[ \sum_{n=1}^{50} 2 = 2 \times 50 = 100 \]

### Step 5: Calculate the second sum

The second sum is a geometric series:
\[ \sum_{n=1}^{50} \frac{1}{2^{n-1}} = 1 + \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2^{49}} \]

Using the formula for the sum of a geometric series:
\[ S = \frac{a(1 - r^n)}{1 - r} \]
where \( a = 1 \), \( r = \frac{1}{2} \), and \( n = 50 \):
\[ S = \frac{1(1 - (1/2)^{50})}{1 - (1/2)} = \frac{1 - \frac{1}{2^{50}}}{\frac{1}{2}} = 2(1 - \frac{1}{2^{50}}) = 2 - \frac{2}{2^{50}} = 2 - \frac{1}{2^{49}} \]

### Step 6: Combine the results

Now substituting back into the equation for \( S_{50} \):
\[ S_{50} = 100 - \left( 2 - \frac{1}{2^{49}} \right) = 100 - 2 + \frac{1}{2^{49}} = 98 + \frac{1}{2^{49}} \]

### Step 7: Identify \( p \) and \( q \)

From the expression:
\[ S_{50} = 98 + \frac{1}{2^{49}} \]
we can identify:
- \( p = 98 \)
- \( q = 49 \)

### Step 8: Calculate \( p + q \)

Finally, we find:
\[ p + q = 98 + 49 = 147 \]

Thus, the value of \( p + q \) is:
\[ \boxed{147} \]