Evaluate the Summation sum from k=1 to 42 of k
The question is asking for an assessment of a mathematical problem involving summation (also known as sigma notation). The task is to calculate the total sum by adding up all the integer values of 'k' starting from 1 and going up to 42. In other words, you would need to find the result of 1 + 2 + 3 + ... + 42. This type of question is testing your ability to apply the concept of summation and possibly to utilize the formula for the sum of the first n natural numbers.
$\sum_{k = 1}^{42} k$
Answer
Solution:
Step 1: Apply the arithmetic series sum formula: $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$
Step 2: Insert the given number into the formula: $\frac{42(42 + 1)}{2}$
Step 3: Begin simplification.
Step 3.1: Identify and remove common factors.
Step 3.1.1: Extract the factor of $2$ from $42(42 + 1)$: $\frac{2(21(42 + 1))}{2}$
Step 3.1.2: Eliminate identical factors.
Step 3.1.2.1: Separate the factor of $2$ from the denominator: $\frac{2(21(42 + 1))}{2(1)}$
Step 3.1.2.2: Cancel out the factor of $2$: $\frac{\cancel{2}(21(42 + 1))}{\cancel{2} \cdot 1}$
Step 3.1.2.3: Reword the fraction: $\frac{21(42 + 1)}{1}$
Step 3.1.2.4: Divide $21(42 + 1)$ by $1$: $21(42 + 1)$
Step 3.2: Finalize the simplification.
Step 3.2.1: Combine $42$ and $1$: $21 \cdot 43$
Step 3.2.2: Calculate the product of $21$ and $43$: $903$
Knowledge Notes:
The problem involves finding the sum of the first $n$ natural numbers, which is a common problem in arithmetic series. The knowledge points related to solving this problem include:
Arithmetic Series: An arithmetic series is the sum of terms in an arithmetic sequence, where each term after the first is obtained by adding a constant difference to the previous term.
Summation Formula: The sum of the first $n$ natural numbers is given by the formula $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$. This is a specific case of the arithmetic series sum formula.
Simplification: The process of simplification involves performing operations to make an expression easier to understand or work with. This can include factoring out common terms, canceling out identical factors in the numerator and denominator, and performing basic arithmetic operations like addition and multiplication.
Factoring: Factoring is the process of breaking down an expression into a product of simpler factors. It is often used to simplify expressions and solve equations.
Cancellation: When a factor appears in both the numerator and denominator of a fraction, it can be canceled out, simplifying the fraction to an equivalent form with lower terms.
Understanding these concepts is crucial for solving problems related to arithmetic series and for performing algebraic manipulations in general.
