Find the sum of the infinite G.P. `7,-1,1//7,-1//49 . . . . . . . .oo`.

Options

A. `49//36`
B. `-7//6`
C. `-49//8`
D. None of these

Answer

To find the sum of the infinite geometric progression (G.P.) given by the terms \(7, -1, \frac{1}{7}, -\frac{1}{49}, \ldots\), we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \(a\) of the G.P. is:
\[ a = 7 \]

To find the common ratio \(r\), we can take the ratio of the second term to the first term:
\[ r = \frac{-1}{7} \]

### Step 2: Check if the common ratio is valid for an infinite G.P.
For an infinite G.P. to converge, the absolute value of the common ratio must be less than 1:
\[ |r| = \left| \frac{-1}{7} \right| = \frac{1}{7} < 1 \]
Since this condition is satisfied, we can proceed to find the sum.

### Step 3: Use the formula for the sum of an infinite G.P.
The formula for the sum \(S\) of an infinite G.P. is given by:
\[ S = \frac{a}{1 - r} \]
Substituting the values of \(a\) and \(r\):
\[ S = \frac{7}{1 - \left(-\frac{1}{7}\right)} = \frac{7}{1 + \frac{1}{7}} = \frac{7}{\frac{8}{7}} = 7 \cdot \frac{7}{8} = \frac{49}{8} \]

### Step 4: Conclusion
Thus, the sum of the infinite G.P. is:
\[ \boxed{\frac{49}{8}} \]