Evaluate the Summation sum from k=5 to 4 of 10k
The problem is asking for the evaluation of a summation (also known as a series), where you sum up the values generated by the expression "10k" as the variable k takes on values in an integer sequence from 5 to 4. This summation would involve adding together the results of substituting k with each integer in the range from the lower bound (5) to the upper bound (4). However, since the upper bound (4) is less than the lower bound (5), this indicates a potential mistake in the problem as normally the summation would be from a lower number to a higher number. The question is to carry out the mathematical operation of summing the products of 10 and k for the given range, if the range were corrected.
$\sum_{k = 5}^{4} 10 k$
Answer
Solution:
Step 1:
When the lower limit of a summation is larger than the upper limit, we are dealing with a situation where the sum is over no terms. This type of summation is known as an empty sum. By definition, the value of an empty sum is zero. Therefore, the summation of \(10k\) from \(k=5\) to \(4\) is \(0\).
Knowledge Notes:
The problem involves evaluating a summation, which is a mathematical notation used to represent the addition of a sequence of numbers. The general form of a summation is:
\[ \sum_{i=m}^{n} a_i \]
where \(i\) is the index of summation, \(m\) is the lower limit, \(n\) is the upper limit, and \(a_i\) is the expression involving \(i\) that is to be summed.
In this case, the expression to be summed is \(10k\), and the summation is from \(k=5\) to \(4\). However, since the lower limit \(5\) is greater than the upper limit \(4\), there are no values of \(k\) to plug into the expression \(10k\). This results in an empty summation.
An empty summation is a concept in mathematics where the sum is taken over an empty set of indices. By convention, the sum over an empty set is defined to be zero. This is because adding up no numbers at all is equivalent to having a total of zero.
In summary, when faced with an empty summation, one should recognize that the result is always zero, regardless of the expression being summed. This is an important concept in the study of series and summations in mathematics.
