Evaluate the Summation sum from k=0 to 3 of k^2+4

Solution:

Step 1:

Write out the terms of the summation for $k = 0$ to $k = 3$: $0^2 + 4, 1^2 + 4, 2^2 + 4, 3^2 + 4$.

Step 2:

Perform the calculations.

Step 2.1:

Calculate $0^2$: $0 + 4, 1^2 + 4, 2^2 + 4, 3^2 + 4$.

Step 2.2:

Combine $0$ and $4$: $4, 1^2 + 4, 2^2 + 4, 3^2 + 4$.

Step 2.3:

Calculate $1^2$: $4, 1 + 4, 2^2 + 4, 3^2 + 4$.

Step 2.4:

Combine $1$ and $4$: $4, 5, 2^2 + 4, 3^2 + 4$.

Step 2.5:

Combine $4$ and $5$: $9, 2^2 + 4, 3^2 + 4$.

Step 2.6:

Calculate $2^2$: $9, 4 + 4, 3^2 + 4$.

Step 2.7:

Combine $4$ and $4$: $9, 8, 3^2 + 4$.

Step 2.8:

Combine $9$ and $8$: $17, 3^2 + 4$.

Step 2.9:

Calculate $3^2$: $17, 9 + 4$.

Step 2.10:

Combine $9$ and $4$: $17, 13$.

Step 2.11:

Combine $17$ and $13$: $30$.

The sum of the series is $\boxed{30}$.

Knowledge Notes:

The problem involves evaluating a finite summation, which is a process of adding up the values of a function at discrete points. The function in this case is $k^2 + 4$, and we are summing over the integer values of $k$ from $0$ to $3$.

Key knowledge points include:

1. Summation Notation: The sigma notation $\sum$ represents the sum of a sequence of numbers. The expression under the sigma symbol tells us what to add, and the limits above and below tell us where to start and stop.

2. Exponents: Raising a number to the power of $2$ (squared) means multiplying the number by itself.

3. Arithmetic Operations: Basic operations such as addition are used to combine terms in the series.

4. Sequence and Series: A sequence is a list of numbers in a specific order, and a series is the sum of the terms of a sequence.

The problem-solving process involves expanding the series, performing exponentiation, and then successive additions to find the total sum.