Sum the series to infinity :
16 ,-8,4 , .....

Answer

To find the sum of the series to infinity: 16, -8, 4, ..., we can follow these steps:

### Step 1: Identify the series
We start with the series:
\[ S = 16 - 8 + 4 - ... \]

### Step 2: Check if the series is a geometric progression (GP)
To determine if this series is a geometric progression, we need to find the common ratio (r) between the terms.

- The first term \( a_1 = 16 \)
- The second term \( a_2 = -8 \)
- The third term \( a_3 = 4 \)

Now, calculate the common ratio:
\[ r = \frac{a_2}{a_1} = \frac{-8}{16} = -\frac{1}{2} \]
\[ r = \frac{a_3}{a_2} = \frac{4}{-8} = -\frac{1}{2} \]

Since both ratios are equal, the series is indeed a geometric progression with:
- First term \( a = 16 \)
- Common ratio \( r = -\frac{1}{2} \)

### Step 3: Use the formula for the sum of an infinite GP
The formula for the sum \( S \) of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
where \( |r| < 1 \). In this case, \( |r| = \frac{1}{2} < 1 \), so we can use the formula.

### Step 4: Substitute the values into the formula
Substituting \( a = 16 \) and \( r = -\frac{1}{2} \) into the formula:
\[ S = \frac{16}{1 - (-\frac{1}{2})} \]
\[ S = \frac{16}{1 + \frac{1}{2}} = \frac{16}{\frac{3}{2}} = 16 \times \frac{2}{3} = \frac{32}{3} \]

### Final Answer
Thus, the sum of the series to infinity is:
\[ S = \frac{32}{3} \]

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