Find the sum of infinite term of the following series : `16+8+4…oo`

Explanation

To find the sum of the infinite series \(16 + 8 + 4 + \ldots\), we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \(a\) of the series is \(16\).

To find the common ratio \(r\), we can divide the second term by the first term:
\[ r = \frac{8}{16} = \frac{1}{2} \]

We can also verify this by dividing the third term by the second term:
\[ r = \frac{4}{8} = \frac{1}{2} \]

Thus, the common ratio \(r\) is \(\frac{1}{2}\).

### Step 2: Check if the series is a geometric series
Since we have a constant ratio between consecutive terms, this series is a geometric progression (GP).

### Step 3: Use the formula for the sum of an infinite geometric series
The formula for the sum \(S\) of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
where \(a\) is the first term and \(r\) is the common ratio.

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

### Step 5: Conclusion
Therefore, the sum of the infinite series \(16 + 8 + 4 + \ldots\) is \(32\).

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