Evaluate the Summation sum from k=1 to 10 of 3k+1

Solution:

Step 1

Decompose the given summation into two separate summations using the distributive property of summation.

$$\sum_{k = 1}^{10} (3k + 1) = 3\sum_{k = 1}^{10} k + \sum_{k = 1}^{10} 1$$

Step 2

Calculate the summation $3 \sum_{k = 1}^{10} k$.

Step 2.1

Apply the arithmetic series formula:

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

Step 2.2

Insert the upper limit of the summation into the formula and multiply by 3.

$$3 \left( \frac{10(10 + 1)}{2} \right)$$

Step 2.3

Perform the arithmetic operations.

Step 2.3.1

Sum the numbers 10 and 1.

$$3 \frac{10 \cdot 11}{2}$$

Step 2.3.2

Calculate the product of 10 and 11.

$$3 \left( \frac{110}{2} \right)$$

Step 2.3.3

Divide 110 by 2.

$$3 \cdot 55$$

Step 2.3.4

Multiply 3 by 55.

$$165$$

Step 3

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

Step 3.1

Use the formula for the summation of a constant term:

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

Step 3.2

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

$$(1)(10)$$

Step 3.3

Multiply the constant 1 by 10.

$$10$$

Step 4

Combine the results from the two summations.

$$165 + 10$$

Step 5

Add the two results together.

$$175$$

Knowledge Notes:

The problem involves evaluating a finite summation of a linear expression. The process requires knowledge of summation properties and formulas for arithmetic series. Here are the relevant knowledge points:

1. Summation Properties: Summation properties allow us to break down complex summations into simpler components. For example, the distributive property enables us to separate a summation of terms into individual summations.

2. Arithmetic Series 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 formula is derived from the fact that the sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms.

3. Summation of a Constant: The sum of a constant $c$ over $n$ terms is simply $cn$ because each term in the summation is the same.

4. Arithmetic Operations: Basic arithmetic operations such as addition, multiplication, and division are used to simplify expressions within the summation.

5. Combining Summations: After evaluating individual summations, their results are combined (usually by addition) to find the total sum.

By understanding and applying these concepts, one can solve summation problems involving linear expressions and constants.