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Summation notation, also known as sigma notation, provides a concise and standardized way to represent the sum of a series of terms. The Greek letter Σ (sigma) is used to express this kind of summation. For example, \(\sum_{i=1}^{2} 5(2)^{i}\) tells you to sum the terms of the expression \(5(2)^{i}\) as i goes from 1 to 2. Summation notation includes several important components:
The lower bound: This tells you where to start your summation. In our example, it is i=1.
The upper bound: This indicates where to end the summation. In the example, it is i=2.
The expression to be summed: Here, it is \(5(2)^{i}\).

To evaluate the series, you substitute each integer value from the lower bound to the upper bound into the expression and sum the results. This offers a clear and streamlined method for dealing with series, making complex problems easier to handle.

A geometric series is a series of terms where each term is found by multiplying the previous term by a constant. This constant is known as the common ratio. For example, the series \(5, 10, 20\) is a geometric series with a common ratio of 2 because every term is obtained by multiplying the previous term by 2.

The given problem \(\sum_{i=1}^{2} 5(2)^{i}\) is a geometric series since each term follows the form \(ar^{n}\), where a is the initial term and r is the common ratio. For this series:
The initial term (a) is 5.
The common ratio (r) is 2.

A geometric series can be summed using specific formulas, especially if it extends to infinity. However, in our example, since it ends at \(i=2\), you simply calculate and add the individual terms.

Evaluating a series involves several methodical steps to ensure the correct sum of terms. Here's a detailed guide based on the example \(\sum_{i=1}^{2} 5(2)^{i}\):

1. **Identify the series terms**: Determine the expression and the range for summation. Here, \(5(2)^{i}\) and i ranges from 1 to 2.
2. **Calculate the first term**: Substitute the lowest bound into the expression \(5(2)^{i}\). For \(i=1\), you get: \(5(2)^{1} = 10\).
3. **Calculate the second term**: Substitute the next integer within the range. For \(i=2\), you get: \(5(2)^{2} = 20\).
4. **Sum the terms**: Add the calculated values: \(10 + 20 = 30\).

These steps ensure accurate and efficient series evaluation. Practice these steps with different series to build confidence and proficiency in solving similar problems.