Evaluate the Summation sum from k=5 to 4 of 2*5k

Solution:

Step 1: Simplify the given summation expression.

Step 1.1: Perform the multiplication of constants.

Multiply the constant $2$ with the base number $5$ to get $10k$.

Step 1.2: Express the simplified summation.

The summation now reads $\sum _{k=5}^{4}10k$.

Step 2: Evaluate the summation.

Given that the lower limit of the summation ($k=5$) exceeds the upper limit ($k=4$), this implies the summation is over an empty set. Therefore, the summation is equal to $0$.

Knowledge Notes:

The problem involves evaluating a summation, which is a mathematical notation used to represent the addition of a sequence of numbers. The summation has a lower and upper limit, indicating the range of values to sum over. In this case, the summation is written as $\sum _{k=5}^{4}2\cdot 5k$, where $k$ is the variable that takes on values from the lower limit to the upper limit.

1. Multiplication of Constants: When a constant is multiplied by a variable within a summation, the constant can be factored out. In this case, $2$ multiplied by $5$ gives $10$, which is then multiplied by $k$.

2. Empty Summation: A summation is considered empty if the lower limit is greater than the upper limit. By convention, the sum over an empty set is defined to be zero. This is because there are no terms to add together, so the sum is the additive identity, which is $0$.

3. Summation Notation: The summation notation consists of the sigma symbol ($\mathrm{\Sigma }$), the index of summation (in this case, $k$), the lower limit of summation (the starting value of $k$), and the upper limit of summation (the ending value of $k$). The expression to the right of the sigma indicates the terms to be summed.

4. Evaluating Summations: When evaluating summations, it is important to first simplify the expression inside the summation as much as possible. After simplification, one can determine if the summation is over a valid range. If the range is valid, the terms are summed; if not, as in this case, the summation is empty and therefore equals zero.