Evaluate the Summation sum from k=1 to 3 of k^3-6
The question asks to calculate the total of a specific mathematical series. This series is defined by the expression k^3 - 6, where k takes on integer values starting from 1 and ending at 3. Specifically, the task is to find the sum of the values of this expression when k is 1, 2, and 3, respectively. This involves evaluating the expression for each value of k, and then adding all of these values together to find the total sum.
$\sum_{k = 1}^{3} k^{3} - 6$
Answer
Solution:
Step 1:
Write out the summation term for each integer from $k=1$ to $k=3$: $1^3 - 6 + 2^3 - 6 + 3^3 - 6$.
Step 2:
Proceed to evaluate the expression step by step.
Step 2.1:
Calculate the cube of $1$: $1^3 = 1$. The expression becomes $1 - 6 + 2^3 - 6 + 3^3 - 6$.
Step 2.2:
Subtract $6$ from the cube of $1$: $1 - 6 = -5$. The expression now reads $-5 + 2^3 - 6 + 3^3 - 6$.
Step 2.3:
Calculate the cube of $2$: $2^3 = 8$. Update the expression to $-5 + 8 - 6 + 3^3 - 6$.
Step 2.4:
Subtract $6$ from the cube of $2$: $8 - 6 = 2$. The expression simplifies to $-5 + 2 + 3^3 - 6$.
Step 2.5:
Combine the results so far: $-5 + 2 = -3$. The expression is now $-3 + 3^3 - 6$.
Step 2.6:
Calculate the cube of $3$: $3^3 = 27$. Incorporate this to get $-3 + 27 - 6$.
Step 2.7:
Subtract $6$ from the cube of $3$: $27 - 6 = 21$. The expression becomes $-3 + 21$.
Step 2.8:
Add the remaining terms: $-3 + 21 = 18$.
The final result of the summation is $18$.
Knowledge Notes:
The problem involves evaluating a finite summation, which is a process of adding up a sequence of terms generated by substituting a range of integers into a given function. In this case, the function is $k^3 - 6$, and the range of integers is from $k=1$ to $k=3$.
To solve this, we follow these steps:
Expansion: We expand the summation by computing the function for each integer within the range, creating a series of terms to be added together.
Simplification: We simplify the series by performing the arithmetic operations in sequence, which includes:
Raising the integer to the power of three (cubing the number).
Subtracting six from the cubed number.
Adding or subtracting the results as we move through the series.
Arithmetic Operations: The operations involved are exponentiation (raising to a power), subtraction, and addition.
Final Summation: After simplifying all terms, we add them together to find the final sum.
In this problem, we use basic arithmetic operations and the concept of summation of a series. Each term of the series is evaluated by substituting the value of $k$ into the function $k^3 - 6$ and performing the necessary calculations. The final result is obtained by adding all the simplified terms.
