Evaluate the Summation sum from k=1 to 10 of k+1

Solution:

Step 1:

Decompose the original summation into two separate summations that are easier to compute. $$\sum_{k = 1}^{10} (k + 1) = \sum_{k = 1}^{10} k + \sum_{k = 1}^{10} 1$$

Step 2:

Calculate the first summation, $\sum_{k = 1}^{10} k$.

Step 2.1:

Use the arithmetic series summation formula for a linear sequence:

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

Step 2.2:

Insert the upper limit of the summation into the formula:

$$\frac{10(10 + 1)}{2}$$

Step 2.3:

Proceed to simplify the expression.

Step 2.3.1:

Identify and eliminate common factors between numerator and denominator.

Step 2.3.1.1:

Extract the factor of 2 from the numerator:

$$\frac{2(5(10 + 1))}{2}$$

Step 2.3.1.2:

Eliminate the common factors.

Step 2.3.1.2.1:

Extract the factor of 2 from the denominator:

$$\frac{2(5(10 + 1))}{2(1)}$$

Step 2.3.1.2.2:

Cancel out the common factor of 2:

$$\frac{\cancel{2}(5(10 + 1))}{\cancel{2} \cdot 1}$$

Step 2.3.1.2.3:

Rewrite the simplified expression:

$$\frac{5(10 + 1)}{1}$$

Step 2.3.1.2.4:

Divide the numerator by the denominator:

$$5(10 + 1)$$

Step 2.3.2:

Further simplify the expression.

Step 2.3.2.1:

Combine the terms within the parentheses:

$$5 \cdot 11$$

Step 2.3.2.2:

Perform the multiplication:

$$55$$

Step 3:

Compute the second summation, $\sum_{k = 1}^{10} 1$.

Step 3.1:

Apply the constant series summation formula:

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

Step 3.2:

Plug in the constant value and the upper limit of the summation:

$$(1)(10)$$

Step 3.3:

Carry out the multiplication:

$$10$$

Step 4:

Combine the results from the two summations:

$$55 + 10$$

Step 5:

Add the two numbers together to get the final result:

$$65$$

Knowledge Notes:

The problem involves evaluating a finite summation, which is a common operation in mathematics, particularly in calculus and discrete mathematics. The summation is a series where each term is defined by a function of the index variable, in this case, k. The process of solving the problem involves several key knowledge points:

1. Summation Decomposition: A summation of a sequence that can be separated into multiple sequences can be split into separate summations. This is useful when dealing with a sequence that is the sum of a polynomial term and a constant term.

2. Arithmetic Series 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 direct result of the formula for an arithmetic series where the common difference is 1.

3. Constant Series Summation Formula: The sum of a constant c over n terms is simply the product of the constant and the number of terms:

   $$\sum_{k = 1}^{n} c = c \cdot n$$ This is because each term in the series is the same, and adding c to itself n times is equivalent to multiplying c by n.

4. Simplification of Fractions: When simplifying fractions, any common factor in the numerator and denominator can be canceled out. This is a basic principle of arithmetic that simplifies calculations.

5. Basic Arithmetic Operations: The final steps involve performing basic arithmetic operations such as addition and multiplication to arrive at the final result.