Evaluate the Summation sum from k=1 to 10 of k^3-k
The question is asking for the evaluation of a finite summation. Specifically, it requests the summation of the difference of two expressions for each integer value of 'k' starting from 1 up to and including 10. The expressions involved in the difference are 'k cubed' (k^3) and 'k' itself. The task is to find the total sum after computing this difference for each value of 'k' in the specified range.
$\sum_{k = 1}^{10} k^{3} - k$
Answer
Solution:
Step 1: Decompose the given summation into two separate summations that are easier to calculate. $\sum_{k = 1}^{10} k^{3} - k$ can be expressed as $\sum_{k = 1}^{10} k^{3} - \sum_{k = 1}^{10} k$.
Step 2: Calculate the summation $\sum_{k = 1}^{10} k^{3}$.
Step 2.1: Use the formula for the sum of cubes: $\sum_{k = 1}^{n} k^{3} = \left(\frac{n(n + 1)}{2}\right)^{2}$.
Step 2.2: Insert the upper limit of the summation into the formula: $\left(\frac{10(10 + 1)}{2}\right)^{2}$.
Step 2.3: Perform the calculations.
Step 2.3.1: First, work out the numerator.
Step 2.3.1.1: Combine $10$ and $1$: $\left(\frac{10^2 \cdot 11^2}{4}\right)$.
Step 2.3.1.2: Square $10$: $\left(\frac{100 \cdot 11^2}{4}\right)$.
Step 2.3.1.3: Square $11$: $\left(\frac{100 \cdot 121}{4}\right)$.
Step 2.3.2: Now, simplify the entire expression.
Step 2.3.2.1: Multiply $100$ by $121$: $\frac{12100}{4}$.
Step 2.3.2.2: Divide $12100$ by $4$: $3025$.
Step 3: Now, compute the summation $\sum_{k = 1}^{10} k$.
Step 3.1: Apply the formula for the sum of the first $n$ natural numbers: $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$.
Step 3.2: Insert the upper limit of the summation into the formula and include the coefficient: $-1 \cdot \left(\frac{10(10 + 1)}{2}\right)$.
Step 3.3: Carry out the simplification process.
Step 3.3.1: Add $10$ and $1$: $-1 \cdot \frac{10 \cdot 11}{2}$.
Step 3.3.2: Multiply $10$ by $11$: $-1 \cdot \left(\frac{110}{2}\right)$.
Step 3.3.3: Divide $110$ by $2$: $-1 \cdot 55$.
Step 3.3.4: Multiply $-1$ by $55$: $-55$.
Step 4: Combine the results from the two summations: $3025 - 55$.
Step 5: Finally, subtract $55$ from $3025$ to get the answer: $2970$.
Knowledge Notes:
Summation is a mathematical operation that adds up a sequence of numbers, typically expressed as $\sum$.
The sum of the first $n$ natural numbers is given by the formula $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$.
The sum of the cubes of the first $n$ natural numbers is given by the formula $\sum_{k = 1}^{n} k^{3} = \left(\frac{n(n + 1)}{2}\right)^{2}$.
When dealing with summations, it is often useful to break them down into simpler parts that can be calculated using known formulas.
Simplifying expressions involves performing operations like addition, multiplication, and division to reduce the expression to its simplest form.
