Evaluate the Summation sum from k=1 to 250 of 2
The problem is asking to compute the result of a given mathematical expression which is a summation. Specifically, the summation starts with the integer k equal to 1 and ends with k equal to 250. For each value of k within this range, the term 2 (that appears to be the constant being summed) is to be added together. The question requires calculating the total sum after adding the constant term 2 for each of the 250 terms in the sequence.
$\sum_{k = 1}^{250} 2$
Answer
Solution:
Step 1:
Identify the formula for summing a constant value, which is: $\sum_{k = 1}^{n} c = c \cdot n$
Step 2:
Insert the given values into the identified formula: $2 \cdot 250$
Step 3:
Calculate the product of $2$ and $250$: $500$
Knowledge Notes:
The problem requires evaluating a summation where the term being summed is a constant. In mathematics, when you sum a constant $c$ over $n$ terms, the result is simply the constant multiplied by the number of terms. This is because each term in the summation is the same, and adding $c$ to itself $n$ times is equivalent to multiplying $c$ by $n$.
The relevant knowledge points for solving this problem include:
Summation Notation: Summation notation is a way to represent the addition of a sequence of numbers. The symbol $\sum$ denotes the summation.
Constants: A constant is a value that does not change.
Arithmetic Operations: Basic arithmetic operations include addition, subtraction, multiplication, and division. In this case, we use multiplication.
Substitution: Substitution is the process of replacing a variable with its value.
Multiplication of Numbers: Multiplication is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction, and division. The multiplication of two whole numbers results in the total number of objects that are arranged in a rectangular grid.
In this specific problem, we use the knowledge of summation of constants and basic multiplication to find the result. The formula $\sum_{k = 1}^{n} c = c \cdot n$ is applied by substituting $c$ with $2$ and $n$ with $250$, and then multiplying these two values to get the final answer.
