Find the sum to infinity of the following Geometric Progression: `6, 1.2, 0.24, ...`

Explanation

To find the sum to infinity of the given geometric progression (GP) `6, 1.2, 0.24, ...`, we can follow these steps:

### Step 1: Identify the first term (A)
The first term of the GP is given as:
\[ A = 6 \]

### Step 2: Determine the common ratio (R)
To find the common ratio, we can divide the second term by the first term:
\[ R = \frac{1.2}{6} = 0.2 \]

We can also verify this by dividing the third term by the second term:
\[ R = \frac{0.24}{1.2} = 0.2 \]

### Step 3: Check if the series converges
For a GP to have a sum to infinity, the common ratio \( R \) must satisfy the condition \( |R| < 1 \). Since \( R = 0.2 \), which is less than 1, the series converges.

### Step 4: 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 we found:
\[ S = \frac{6}{1 - 0.2} \]

### Step 5: Simplify the expression
Calculating the denominator:
\[ 1 - 0.2 = 0.8 \]
Now substituting this back into the equation:
\[ S = \frac{6}{0.8} \]

### Step 6: Perform the division
To simplify \( \frac{6}{0.8} \):
\[ S = \frac{6 \times 10}{8} = \frac{60}{8} = \frac{15}{2} \]

### Step 7: Convert to decimal
Finally, converting \( \frac{15}{2} \) to decimal:
\[ S = 7.5 \]

### Conclusion
The sum to infinity of the given geometric progression is:
\[ \boxed{7.5} \]

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