Evaluate the Summation sum from k=4 to 5 of k^2-2k
The question is asking for the calculation of a specific mathematical expression called a summation. Specifically, it's requesting the evaluation of the sum of the squares of k (where k is a variable representing integer values in this context) minus two times k, for all the integer values of k starting from 4 up to 5. The operation is to perform the given calculation for each value of k within that range and then add up all resulting values to find the total sum.
$\sum_{k = 4}^{5} k^{2} - 2 k$
Answer
Solution:
Step:1
Write out the terms of the summation for each integer value of $k$ within the given range.
$4^{2} - 2 \cdot 4 + 5^{2} - 2 \cdot 5$
Step:2
Perform the simplification process.
Step:2.1
Calculate the square of $4$.
$16 - 2 \cdot 4 + 5^{2} - 2 \cdot 5$
Step:2.2
Compute the product of $-2$ and $4$.
$16 - 8 + 5^{2} - 2 \cdot 5$
Step:2.3
Execute the subtraction of $8$ from $16$.
$8 + 5^{2} - 2 \cdot 5$
Step:2.4
Find the square of $5$.
$8 + 25 - 2 \cdot 5$
Step:2.5
Determine the product of $-2$ and $5$.
$8 + 25 - 10$
Step:2.6
Carry out the subtraction of $10$ from $25$.
$8 + 15$
Step:2.7
Sum the results of $8$ and $15$.
$23$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a process of adding up the values of a function at discrete points. In this case, the function is $k^2 - 2k$, and we are summing over the integer values of $k$ from $4$ to $5$.
Relevant knowledge points include:
Understanding of summation notation: The summation symbol $\sum$ is followed by an expression representing the terms to be added, with a variable that changes value according to specified limits.
Exponentiation: Raising a number to the power of $2$ (squaring) is a basic arithmetic operation.
Multiplication and subtraction: These are fundamental operations in arithmetic used to simplify expressions.
Sequential computation: Following a step-by-step process to simplify and evaluate an expression.
The solution involves expanding the summation into individual terms, simplifying each term by performing arithmetic operations, and then adding the simplified terms together to find the final result.
