Evaluate the Summation sum from x=1 to 5 of x+4

Solution:

Step 1: Expansion of the Summation

Write out the terms of the summation by substituting each value of $x$ from 1 to 5 into the expression $x + 4$. The expanded series looks like this: $1 + 4 + 2 + 4 + 3 + 4 + 4 + 4 + 5 + 4$.

Step 2: Calculation of the Sum

Add up all the terms in the expanded series to find the sum. The total is $35$.

Knowledge Notes:

The problem involves evaluating a finite summation. A summation is a mathematical notation used to represent the addition of a sequence of numbers. The notation $\sum$ is used to denote the summation, and it is followed by an expression that represents the terms in the series. The variable under the summation sign (in this case, $x$) is called the index of summation, and it takes on values within a specified range.

In this problem, we are asked to evaluate the summation of $x + 4$ as $x$ goes from 1 to 5. This means we need to calculate the sum of the series when $x$ takes on each integer value from 1 to 5.

To solve this, we follow these steps:

1. Expansion: We substitute each value of $x$ into the expression $x + 4$ and write out all the terms. This step is crucial as it lays out all the components that will be summed together.

2. Simplification: After expanding the series, we add all the terms together to get the final sum. Since the series is arithmetic (each term is a constant difference from the previous), we can simply add the terms directly to find the total sum.

In this case, the series expansion is straightforward because the expression $x + 4$ is a simple linear function of $x$. The summation of such a series can be done by direct addition.

The final sum of the series is calculated to be $35$, which is the result of adding all the terms from the expanded series together.