Evaluate the Summation sum from k=2 to 4 of k^3

Solution:

Step:1

List out each term of the summation with the corresponding power of $k$.

$2^3 + 3^3 + 4^3$

Step:2

Perform the calculations.

Step:2.1

Compute $2$ raised to the $3$rd power.

$8 + 3^3 + 4^3$

Step:2.2

Compute $3$ raised to the $3$rd power.

$8 + 27 + 4^3$

Step:2.3

Combine the first two terms.

$35 + 4^3$

Step:2.4

Compute $4$ raised to the $3$rd power.

$35 + 64$

Step:2.5

Sum the final two terms.

$99$

Knowledge Notes:

The problem involves evaluating a finite summation, which is a process of adding up all the terms where the variable $k$ takes on each integer value from the lower limit (in this case, $2$) to the upper limit (in this case, $4$). Each term is the value of $k$ raised to the third power ($k^3$).

To solve this, one needs to understand the following concepts:

1. Summation Notation: The summation notation $\sum$ is used to denote the sum of a sequence of terms. The variable below the summation sign is the index of summation, and the numbers below and above the sign indicate the starting and ending values, respectively.

2. Exponents: Raising a number to the power of $3$ (cubing) means multiplying the number by itself three times.

3. Arithmetic Operations: Basic arithmetic operations, including addition, are used to combine the terms after they have been raised to the appropriate power.

The process involves expanding the series, simplifying each term by raising the base number to the exponent, and then adding the results together to find the sum. The final answer is the sum of the cubes of the integers from $2$ to $4$, which is $99$.