Evaluate the Summation sum from k=2 to 4 of k(k+3)

Solution:

Step 1: Break down the summation expression.

• Step 1.1: Utilize the distributive property to expand the terms: $k \cdot k + k \cdot 3$
• Step 1.2: Perform the multiplication of $k$ with itself: $k^2 + k \cdot 3$
• Step 1.3: Rearrange the terms for clarity: $k^2 + 3k$
• Step 1.4: Rewrite the expanded summation expression: $\sum_{k = 2}^{4} k^2 + 3k$

Step 2: Calculate the series for each individual $k$ value.

• Compute the expression for $k=2$, $k=3$, and $k=4$: $2^2 + 3 \cdot 2$, $3^2 + 3 \cdot 3$, and $4^2 + 3 \cdot 4$

Step 3: Simplify the series.

• Step 3.1: Square the number $2$: $4 + 3 \cdot 2 + 9 + 3 \cdot 3 + 16 + 3 \cdot 4$
• Step 3.2: Multiply $3$ by $2$: $4 + 6 + 9 + 9 + 16 + 12$
• Step 3.3: Combine $4$ and $6$: $10 + 9 + 9 + 16 + 12$
• Step 3.4: Square the number $3$: $10 + 9 + 9 + 16 + 12$
• Step 3.5: Multiply $3$ by $3$: $10 + 9 + 9 + 16 + 12$
• Step 3.6: Add $9$ and $9$: $10 + 18 + 16 + 12$
• Step 3.7: Combine $10$ and $18$: $28 + 16 + 12$
• Step 3.8: Square the number $4$: $28 + 16 + 12$
• Step 3.9: Multiply $3$ by $4$: $28 + 16 + 12$
• Step 3.10: Add $16$ and $12$: $28 + 28$
• Step 3.11: Combine $28$ and $28$: $56$

The final result is $56$.

Knowledge Notes:

The problem involves evaluating a finite summation, which is a common mathematical operation used to add a sequence of numbers that follow a specific pattern. The steps taken to solve the problem include:

1. Distributive Property: This property allows us to multiply a sum by multiplying each addend separately and then add the products. In this case, it is used to expand $k(k+3)$ into $k^2 + 3k$.

2. Arithmetic Operations: Basic arithmetic operations such as addition, multiplication, and exponentiation are used to simplify the expression for each term in the series.

3. Finite Summation: The concept of finite summation involves adding up all the values of a function at discrete points within a certain range. In this problem, the summation is from $k=2$ to $k=4$.

4. Simplification: The process of simplifying involves performing arithmetic operations in the correct order (following the order of operations) to reduce the expression to a single numerical value.

5. Exponentiation: This operation involves raising a number to a power, which is a shorthand for repeated multiplication. For example, $2^2$ means $2$ multiplied by itself, which equals $4$.

6. Combining Like Terms: When simplifying expressions, terms that have the same variables raised to the same power can be combined by adding or subtracting their coefficients.

By following these steps and applying the relevant mathematical rules, the summation can be evaluated to obtain the final result.