Evaluate the Summation sum from k=1 to 17 of 4k+4

Solution:

Step 1: Decompose the given summation

Break down the original summation into two separate summations that are easier to manage.

$$\sum_{k=1}^{17} (4k + 4) = 4\sum_{k=1}^{17} k + \sum_{k=1}^{17} 4$$

Step 2: Calculate the first summation

Compute the summation that involves the variable $k$.

Step 2.1: Use the arithmetic series formula

The sum of the first $n$ natural numbers is given by:

$$\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}$$

Step 2.2: Apply the formula with $n=17$

Insert the upper limit of the summation into the formula and include the coefficient $4$.

$$4 \left( \frac{17(17 + 1)}{2} \right)$$

Step 2.3: Simplify the expression

Step 2.3.1: Perform the addition inside the parentheses

Add together $17$ and $1$.

$$4 \left( \frac{17 \cdot 18}{2} \right)$$

Step 2.3.2: Simplify by reducing the fraction

Eliminate the common factor between the numerator and the coefficient.

Step 2.3.2.1: Factor out a $2$ from $4$

$$2(2) \left( \frac{306}{2} \right)$$

Step 2.3.2.2: Cancel out the $2$s

$$\cancel{2} \cdot 2 \left( \frac{306}{\cancel{2}} \right)$$

Step 2.3.2.3: Finalize the simplification

$$2 \cdot 306$$

Step 2.3.3: Multiply to find the result

$$612$$

Step 3: Calculate the second summation

Evaluate the summation of the constant term.

Step 3.1: Utilize the constant series formula

The sum of a constant $c$ over $n$ terms is:

$$\sum_{k=1}^{n} c = cn$$

Step 3.2: Implement the formula with $c=4$ and $n=17$

$$4(17)$$

Step 3.3: Compute the product

$$68$$

Step 4: Combine the results of both summations

$$612 + 68$$

Step 5: Add the two sums to get the final answer

$$680$$

Knowledge Notes:

The problem requires evaluating a summation, which is a common operation in mathematics, particularly in algebra and calculus. The summation in question is a finite arithmetic series. The knowledge points involved in solving this problem include:

1. Summation Notation: The use of the sigma notation ($\sum$) to represent the sum of a sequence of terms.

2. Arithmetic Series: A sequence of numbers in which each term after the first is obtained by adding a constant to the previous term.

3. Summation Formulas:

   • The sum of the first $n$ natural numbers: $\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}$.

   • The sum of a constant over $n$ terms: $\sum_{k=1}^{n} c = cn$.

4. Distributive Property: This property is used to break down the original summation into two parts that can be handled separately.

5. Simplification: The process of reducing expressions to their simplest form, often by canceling common factors or combining like terms.

6. Arithmetic Operations: Basic operations such as addition, multiplication, and division are used to evaluate the summations.

Understanding these concepts is essential for solving summation problems efficiently and correctly.