Evaluate the Summation sum from k=1 to 500 of 7
The given problem is asking for the calculation of a mathematical summation. Specifically, the summation inquires about the total value obtained when adding the constant number 7 repeatedly for each integral value of k starting at 1 and ending at 500. The task involves finding the sum of 500 terms, each of which has the value of 7.
$\sum_{k = 1}^{500} 7$
Answer
Solution:
Step 1:
The summation of a constant $c$ repeated $n$ times is given by the equation: $\sum_{k = 1}^{n} c = cn$
Step 2:
Insert the given numbers into the equation: $7 \cdot 500$
Step 3:
Calculate the product of $7$ and $500$: $3500$
Knowledge Notes:
The problem involves evaluating a simple summation where the term being summed is a constant. In such cases, the summation process is straightforward because adding the same number $c$ a total of $n$ times is equivalent to multiplying that number by $n$. The relevant formula for this type of summation is $\sum_{k = 1}^{n} c = cn$. This formula is a direct consequence of the definition of summation and the distributive property of multiplication over addition.
To solve the problem, we follow these steps:
Recognize that the summation involves a constant value, which simplifies the process.
Apply the summation formula for a constant, which is the constant value multiplied by the number of terms in the summation.
Perform the multiplication to find the sum.
In this specific case, the constant value $c$ is $7$, and the number of terms $n$ is $500$. By applying the formula, we multiply $7$ by $500$ to get the final result, $3500$. This process demonstrates a fundamental principle in summation, particularly when dealing with constant terms.
