Evaluate the Summation sum from i=1 to 6 of 5
The question is asking for the calculation of a summation, which is a mathematical way to add up a series of numbers. Specifically, you are instructed to find the sum of the constant value 5 as it is repeated for each value of 'i' starting from 'i=1' and ending with 'i=6'. This is not asking you to multiply, but rather to repeatedly add the number 5 to itself six times, since the index 'i' ranges from 1 through 6.
$\sum_{i = 1}^{6} 5$
Answer
Solution:
Step 1:
Identify the formula for the sum of a constant over a range: $\sum_{i = 1}^{n} a = a \cdot n$
Step 2:
Plug in the given constant and the upper limit of the summation into the formula: $5 \cdot 6$
Step 3:
Calculate the product of the constant and the upper limit: $5 \times 6 = 30$
Knowledge Notes:
The problem at hand involves evaluating the summation of a constant number over a finite sequence of integers. The key points to understand here are:
Summation Notation: Summation is often represented using the Greek letter Sigma ($\Sigma$) and is a way to add up a series of numbers. The notation includes an expression to sum, a variable that represents the index of summation, a lower bound, and an upper bound.
Summation of a Constant: When the summation involves a constant (a number that does not depend on the index of summation), the result is simply the constant multiplied by the number of terms in the summation. This is because adding the same number $c$ to itself $n$ times is equivalent to multiplying $c$ by $n$.
Arithmetic Operations: The problem requires basic arithmetic operations, specifically multiplication. Multiplying a number by an integer simply adds the number to itself that many times.
Substitution: The process of solving the problem involves substituting the known values into the summation formula. This is a common technique in mathematics where known values replace the variables in an equation or expression.
Formula Application: The formula for the sum of a constant is derived from the concept of arithmetic series and is a fundamental result in algebra.
In this specific problem, the constant is $5$, and the number of terms ($n$) is $6$, since the summation is from $i=1$ to $i=6$. Applying the formula, the sum is $5 \cdot 6 = 30$.
