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 find the sum of the first 50 terms of the series \(1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \ldots\), we can start by identifying a pattern in the series.

### Step 1: Identify the general term of the series
The numerators of the series appear to follow a pattern:
- The first term is \(1 = 2^1 - 1\)
- The second term is \(3 = 2^2 - 1\)
- The third term is \(7 = 2^3 - 1\)
- The fourth term is \(15 = 2^4 - 1\)
- The fifth term is \(31 = 2^5 - 1\)

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

### Step 2: Rewrite the series
We can rewrite the series using the general term:
\[ S_n = \sum_{n=1}^{50} a_n = \sum_{n=1}^{50} \left( \frac{2^n - 1}{2^{n-1}} \right) = \sum_{n=1}^{50} \left( 2 - \frac{1}{2^{n-1}} \right) \]

### Step 3: Split the sum
Now, we can separate the sum into two parts:
\[ S_n = \sum_{n=1}^{50} 2 - \sum_{n=1}^{50} \frac{1}{2^{n-1}} \]
The first sum is straightforward:
\[ \sum_{n=1}^{50} 2 = 2 \times 50 = 100 \]

### Step 4: Calculate the second sum
The second sum is a geometric series:
\[ \sum_{n=1}^{50} \frac{1}{2^{n-1}} = 1 + \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2^{49}} \]
This is a geometric series with first term \(a = 1\) and common ratio \(r = \frac{1}{2}\). The sum of the first \(n\) terms of a geometric series is given by:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
Substituting in our values:
\[ S_{50} = \frac{1(1 - (1/2)^{50})}{1 - 1/2} = \frac{1 - \frac{1}{2^{50}}}{\frac{1}{2}} = 2(1 - \frac{1}{2^{50}}) = 2 - \frac{2}{2^{50}} = 2 - \frac{1}{2^{49}} \]

### Step 5: Combine the results
Now substituting back into our expression for \(S_n\):
\[ S_n = 100 - \left(2 - \frac{1}{2^{49}}\right) = 100 - 2 + \frac{1}{2^{49}} = 98 + \frac{1}{2^{49}} \]

### Step 6: Identify \(p\) and \(q\)
From the final expression, we can see that:
\[ S_n = 98 + \frac{1}{2^{49}} \]
Thus, \(p = 98\) and \(q = 49\).

### Step 7: Calculate \(p + q\)
Now, we find:
\[ p + q = 98 + 49 = 147 \]

### Final Answer
The value of \(p + q\) is \(147\).