Find the sum to infinity of the following Geometric Progression:`(-3)/4,3/(16),(-3)/(64),...`

Options

A. `-3/5`
B. `-2/5`
C. `3/5`
D. `2/5`

Explanation

To find the sum to infinity of the given geometric progression (GP):

\[ \text{GP: } -\frac{3}{4}, \frac{3}{16}, -\frac{3}{64}, \ldots \]

**Step 1: Identify the first term (a) and the common ratio (r)**

The first term \( a \) of the GP is:

\[ a = -\frac{3}{4} \]

To find the common ratio \( r \), we can divide the second term by the first term:

\[ r = \frac{\frac{3}{16}}{-\frac{3}{4}} = \frac{3}{16} \times -\frac{4}{3} = -\frac{4}{16} = -\frac{1}{4} \]

**Step 2: Check the condition for the sum to infinity**

The sum to infinity of a geometric series exists if the absolute value of the common ratio is less than 1:

\[ |r| = \left| -\frac{1}{4} \right| = \frac{1}{4} < 1 \]

Since this condition is satisfied, we can proceed to calculate the sum to infinity.

**Step 3: Use the formula for the sum to infinity**

The formula for the sum to infinity \( S \) of a geometric series is given by:

\[ S = \frac{a}{1 - r} \]

Substituting the values of \( a \) and \( r \):

\[ S = \frac{-\frac{3}{4}}{1 - \left(-\frac{1}{4}\right)} = \frac{-\frac{3}{4}}{1 + \frac{1}{4}} = \frac{-\frac{3}{4}}{\frac{5}{4}} \]

**Step 4: Simplify the expression**

To simplify:

\[ S = -\frac{3}{4} \times \frac{4}{5} = -\frac{3}{5} \]

Thus, the sum to infinity of the given geometric progression is:

\[ \boxed{-\frac{3}{5}} \]

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