Find the sum to infinity of the G.P. `-3/4,3/16,-3/64`,………..

Answer

To find the sum to infinity of the given geometric progression (G.P.) \(-\frac{3}{4}, \frac{3}{16}, -\frac{3}{64}, \ldots\), we will follow these steps:

### Step 1: Identify the first term (a) and the common ratio (r)
The first term \(a_1\) of the G.P. is:
\[ a_1 = -\frac{3}{4} \]

To find the common ratio \(r\), we divide the second term by the first term:
\[ r = \frac{\text{second term}}{\text{first term}} = \frac{\frac{3}{16}}{-\frac{3}{4}} = \frac{3}{16} \times -\frac{4}{3} = -\frac{1}{4} \]

### Step 2: Check the condition for the sum to infinity
The sum to infinity of a G.P. can be calculated 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 \(S_\infty\) of a G.P. is given by:
\[ S_\infty = \frac{a_1}{1 - r} \]
Substituting the values of \(a_1\) and \(r\):
\[ S_\infty = \frac{-\frac{3}{4}}{1 - \left(-\frac{1}{4}\right)} = \frac{-\frac{3}{4}}{1 + \frac{1}{4}} = \frac{-\frac{3}{4}}{\frac{5}{4}} \]

### Step 4: Simplify the expression
To simplify:
\[ S_\infty = -\frac{3}{4} \times \frac{4}{5} = -\frac{3}{5} \]

### Final Answer
Thus, the sum to infinity of the given G.P. is:
\[ S_\infty = -\frac{3}{5} \]

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