Sum of infinity of sequence 5, `(5)/(3),(5)/(9)`,……..________.

Answer

To find the sum of the infinite sequence 5, \( \frac{5}{3} \), \( \frac{5}{9} \), …, we will follow these steps:

### Step 1: Identify the first term and the common ratio
The first term of the sequence \( a \) is 5.

Next, we need to find the common ratio \( r \). The common ratio can be calculated by dividing the second term by the first term:
\[ r = \frac{a_2}{a_1} = \frac{\frac{5}{3}}{5} = \frac{5}{3} \times \frac{1}{5} = \frac{1}{3} \]

### Step 2: Verify the common ratio
To ensure that the sequence is geometric, we should check if the ratio remains constant for the next terms:
\[ r = \frac{a_3}{a_2} = \frac{\frac{5}{9}}{\frac{5}{3}} = \frac{5}{9} \times \frac{3}{5} = \frac{3}{9} = \frac{1}{3} \]
Since both calculations give the same common ratio \( r = \frac{1}{3} \), we confirm that the sequence is geometric.

### 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 the values we found:
\[ S = \frac{5}{1 - \frac{1}{3}} = \frac{5}{\frac{2}{3}} = 5 \times \frac{3}{2} = \frac{15}{2} \]

### Conclusion
The sum of the infinite sequence is:
\[ \frac{15}{2} \]