Find the sum: `sum_(n=1)^(10){(1/2)^(n-1)+(1/5)^(n+1)}dot`

Explanation

To find the sum \( S = \sum_{n=1}^{10} \left( \left( \frac{1}{2} \right)^{n-1} + \left( \frac{1}{5} \right)^{n+1} \right) \), we can break this into two separate sums:

1. \( S_1 = \sum_{n=1}^{10} \left( \frac{1}{2} \right)^{n-1} \)
2. \( S_2 = \sum_{n=1}^{10} \left( \frac{1}{5} \right)^{n+1} \)

### Step 1: Calculate \( S_1 \)

The first sum \( S_1 \) is a geometric series where:
- The first term \( a = 1 \) (when \( n=1 \), \( \left( \frac{1}{2} \right)^{0} = 1 \))
- The common ratio \( r = \frac{1}{2} \)
- The number of terms \( n = 10 \)

The formula for the sum of the first \( n \) terms of a geometric series is:
\[ S_n = a \frac{1 - r^n}{1 - r} \]

Substituting the values:
\[ S_1 = 1 \cdot \frac{1 - \left( \frac{1}{2} \right)^{10}}{1 - \frac{1}{2}} = \frac{1 - \left( \frac{1}{2} \right)^{10}}{\frac{1}{2}} = 2 \left( 1 - \frac{1}{1024} \right) \]
\[ S_1 = 2 \left( \frac{1023}{1024} \right) = \frac{2046}{1024} = \frac{1023}{512} \]

### Step 2: Calculate \( S_2 \)

The second sum \( S_2 \) can be rewritten as:
\[ S_2 = \sum_{n=1}^{10} \left( \frac{1}{5} \right)^{n+1} = \sum_{m=3}^{12} \left( \frac{1}{5} \right)^{m} \quad \text{(where \( m = n + 2 \))} \]
This is also a geometric series where:
- The first term \( a = \left( \frac{1}{5} \right)^{3} = \frac{1}{125} \)
- The common ratio \( r = \frac{1}{5} \)
- The number of terms \( n = 10 \)

Using the sum formula:
\[ S_2 = \frac{1/125 \cdot (1 - (1/5)^{10})}{1 - 1/5} = \frac{\frac{1}{125} \cdot (1 - \frac{1}{5^{10}})}{\frac{4}{5}} = \frac{5}{500} (1 - \frac{1}{9765625}) = \frac{1}{100} (1 - \frac{1}{9765625}) \]

Calculating this gives:
\[ S_2 = \frac{1}{100} \cdot \left( \frac{9765624}{9765625} \right) = \frac{9765624}{976562500} \]

### Step 3: Combine \( S_1 \) and \( S_2 \)

Now we can combine both sums:
\[ S = S_1 + S_2 = \frac{1023}{512} + \frac{9765624}{976562500} \]

To add these fractions, we need a common denominator. The least common multiple of \( 512 \) and \( 976562500 \) can be calculated, but for simplicity, we can convert both fractions to a decimal or find a common denominator.

### Final Answer

The final sum \( S \) can be approximated or calculated exactly based on the above fractions.