Evaluate the Summation sum from k=4 to 5 of k^2+2k

Solution:

Step 1:

Write out the terms of the series for each value of $k$: $4^2+2\cdot 4+5^2+2\cdot 5$.

Step 2:

Perform the simplification.

Step 2.1:

Calculate $4$ squared: $16+2\cdot 4+5^2+2\cdot 5$.

Step 2.2:

Compute $2$ times $4$: $16+8+5^2+2\cdot 5$.

Step 2.3:

Combine $16$ and $8$: $24+5^2+2\cdot 5$.

Step 2.4:

Calculate $5$ squared: $24+25+2\cdot 5$.

Step 2.5:

Compute $2$ times $5$: $24+25+10$.

Step 2.6:

Combine $25$ and $10$: $24+35$.

Step 2.7:

Add $24$ and $35$ together: $59$.

The final result is $59$.

Knowledge Notes:

To solve the given problem, we need to understand the concept of summation and basic arithmetic operations. The summation notation $\sum$ represents the sum of a sequence of numbers, each defined by an expression involving a variable that takes on successive integer values within a specified range. In this problem, we are summing the expression $k^2+2k$ for $k$ values from 4 to 5.

Relevant knowledge points include:

1. Summation Notation: Understanding how to interpret and evaluate the summation symbol $\sum$.

2. Exponentiation: Knowing how to raise a number to a power, which in this case involves squaring the numbers 4 and 5.

3. Multiplication: Multiplying numbers, such as multiplying 2 by 4 and 2 by 5.

4. Addition: Adding a sequence of numbers to find the total sum.

5. Arithmetic Operations: Performing the basic operations of addition, multiplication, and exponentiation in the correct order following the rules of arithmetic.

To solve the problem, we expand the series by calculating the expression for each value of $k$ within the given range. We then simplify by performing the arithmetic operations in the correct order: squaring the numbers, multiplying, and then adding them together. The final step is to sum all the terms to get the result.