Evaluate the Summation sum from i=1 to 6 of 8i

Solution:

Step 1:

Extract the constant $8$ from the sum: $8 \sum_{i = 1}^{6} i$

Step 2:

Apply the arithmetic series sum formula: $\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2}$

Step 3:

Insert the upper limit of the summation into the formula, and don't forget to include the constant from step 1: $8 \left( \frac{6(6 + 1)}{2} \right)$

Step 4:

Proceed with the simplification:

Step 4.1:

Begin simplifying the expression within the parentheses.

Step 4.1.1:

Combine $6$ and $1$: $8 \left( \frac{6 \cdot 7}{2} \right)$

Step 4.1.2:

Calculate the product of $6$ and $7$: $8 \left( \frac{42}{2} \right)$

Step 4.2:

Reduce the fraction by eliminating common factors.

Step 4.2.1:

Factor out a $2$ from the $8$: $2 \cdot 4 \left( \frac{42}{2} \right)$

Step 4.2.2:

Eliminate the common factor of $2$: $4 \cdot \frac{42}{1}$

Step 4.2.3:

Present the simplified expression: $4 \cdot 42$

Step 4.3:

Final multiplication: $4 \cdot 42 = 168$

Knowledge Notes:

The problem involves evaluating the sum of a simple arithmetic series. The relevant knowledge points for solving this problem include:

1. Arithmetic Series: An arithmetic series is the sum of the terms of an arithmetic sequence, a sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term.

2. Summation Notation: The summation notation $\sum$ is used to denote the sum of a sequence of numbers. The variable $i$ is often used as the index of summation, with lower and upper bounds to indicate the range of values to be summed.

3. Summation Formula for an Arithmetic Series: The sum of the first $n$ natural numbers is given by the formula $\sum_{i = 1}^{n} i = \frac{n(n + 1)}{2}$. This formula is derived from the fact that the sum of an arithmetic series can be found by multiplying the average of the first and last terms by the number of terms.

4. Factoring Constants from Summations: When a constant is multiplied by each term in a summation, it can be factored out of the summation to simplify the expression. This is based on the distributive property of multiplication over addition.

5. Simplification: The process of simplification involves performing arithmetic operations such as addition, multiplication, and division to reduce an expression to its simplest form.

6. Cancellation: When a number appears in both the numerator and denominator of a fraction, it can be cancelled out to simplify the expression. This is based on the property that a number divided by itself equals one.

By applying these concepts, the problem is solved in a step-by-step manner, maintaining the structure and format of the original solution.