Evaluate the Summation sum from j=1 to 10 of 3j+6

Solution:

Step 1: Decompose the given summation

Break down the original summation into two separate summations using the distributive property of summation.

$$\sum_{j = 1}^{10} (3j + 6) = 3\sum_{j = 1}^{10} j + \sum_{j = 1}^{10} 6$$

Step 2: Calculate the first summation $3\sum_{j = 1}^{10} j$

Step 2.1: Apply 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: Plug in the upper limit of the summation and the coefficient

Insert $n=10$ into the formula and multiply by the coefficient $3$:

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

Step 2.3: Perform the arithmetic operations

Step 2.3.1: Add the numbers inside the parentheses

Calculate $10 + 1$ and multiply by $10$:

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

Step 2.3.2: Multiply $10$ by $11$

Find the product of $10$ and $11$:

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

Step 2.3.3: Divide $110$ by $2$

Divide $110$ by $2$ to get $55$:

$$3 \cdot 55$$

Step 2.3.4: Multiply $3$ by $55$

Multiply $3$ by $55$ to obtain the sum of the first part:

$$165$$

Step 3: Calculate the second summation $\sum_{j = 1}^{10} 6$

Step 3.1: Use the formula for the summation of a constant

The sum of a constant $c$ repeated $n$ times is:

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

Step 3.2: Substitute the constant and the number of terms

Insert $c=6$ and $n=10$ into the formula:

$$6 \cdot 10$$

Step 3.3: Multiply $6$ by $10$

Find the product of $6$ and $10$ to get the sum of the second part:

$$60$$

Step 4: Combine the results of both summations

Add the results from Step 2 and Step 3:

$$165 + 60$$

Step 5: Find the final sum

Add $165$ and $60$ to reach the final result:

$$225$$

Knowledge Notes:

The problem involves evaluating a finite summation, which is a common task in algebra and calculus. The summation in question is a linear combination of a variable term and a constant term. The solution involves using properties of summations and formulas for specific types of series.

1. Distributive Property of Summation: This property allows us to split a summation of terms into separate summations. For example, $\sum (a + b) = \sum a + \sum b$.

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 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 $c$ multiplied by $n$, or $\sum_{k = 1}^{n} c = c \cdot n$. This is because adding the same number $n$ times is equivalent to multiplication.

4. Arithmetic Operations: The solution requires basic arithmetic operations such as addition, multiplication, and division to evaluate the summations.

5. Combining Results: After evaluating each summation separately, the results are combined through addition to find the final sum of the original summation.

By understanding these concepts and applying the appropriate formulas, one can solve summation problems efficiently.