Find the sum of first `n` term of a G.P.`1+(1)/(2)+(1)/(4)+(1)/(8)+...`

Options

A. `(1)/(2^(n))`
B. `(1)/(2^(n)) + 1`
C. ` 1 - (1)/(2^(n))`
D. `2(1 - (1)/(2^(n)))`

Answer

To find the sum of the first `n` terms of the given geometric progression (G.P.) \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \), we will follow these steps:

### Step 1: Identify the first term and common ratio
The first term \( a \) of the G.P. is:
\[ a = 1 \]
The common ratio \( r \) can be determined by dividing the second term by the first term:
\[ r = \frac{\frac{1}{2}}{1} = \frac{1}{2} \]

### Step 2: Write 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. is given by:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
where \( a \) is the first term and \( r \) is the common ratio.

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

### Step 4: Simplify the expression
Now, simplify the denominator:
\[ 1 - \frac{1}{2} = \frac{1}{2} \]
Thus, the expression for \( S_n \) becomes:
\[ S_n = \frac{1(1 - (\frac{1}{2})^n)}{\frac{1}{2}} = 2(1 - (\frac{1}{2})^n) \]

### Step 5: Final expression for the sum of the first `n` terms
Therefore, the sum of the first \( n \) terms of the G.P. is:
\[ S_n = 2(1 - (\frac{1}{2})^n) \]

### Summary
The sum of the first \( n \) terms of the G.P. \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \) is:
\[ S_n = 2(1 - (\frac{1}{2})^n) \]