Evaluate the Summation sum from x=1 to 3 of x-1
The question is asking for the evaluation of a mathematical expression that involves a summation. Specifically, the summation is defined with a variable x that takes on integer values starting from 1 and going up to 3. For each value of x in this range, the expression x - 1 gets calculated, and the results are then added together to find the total sum. The request is to compute the final summed value of this sequence.
$\sum_{x = 1}^{3} x - 1$
Answer
Solution:
Step 1:
Write out the terms of the summation for each integer value of $x$ from $1$ to $3$.
$$(1 - 1) + (2 - 1) + (3 - 1)$$
Step 2:
Perform the simplification process.
Step 2.1:
Calculate $1 - 1$.
$$0 + (2 - 1) + (3 - 1)$$
Step 2.2:
Calculate $2 - 1$.
$$0 + 1 + (3 - 1)$$
Step 2.3:
Combine $0$ and $1$.
$$1 + (3 - 1)$$
Step 2.4:
Calculate $3 - 1$.
$$1 + 2$$
Step 2.5:
Add $1$ and $2$ together.
$$3$$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a process of adding up a sequence of numbers generated by a formula that depends on a variable. In this case, the variable $x$ takes on integer values from $1$ to $3$, and the formula for the terms of the sequence is $x - 1$.
To solve the problem, one must:
Expansion of the Series: Write out each term of the series explicitly for every value of $x$ within the given range. This step involves substituting the values of $x$ into the formula to generate the terms of the summation.
Simplification: After expanding the series, the next step is to simplify the expression by performing the arithmetic operations indicated. This typically involves addition, subtraction, multiplication, or division, depending on the formula for the terms of the series.
Combining Like Terms: If the series contains like terms (terms that are the same or can be combined through addition or subtraction), they should be combined to simplify the expression further.
Final Calculation: The last step is to perform any remaining arithmetic operations to arrive at the final sum.
In the context of this problem, the simplification steps involve basic arithmetic: subtraction of $1$ from each value of $x$ and then adding the results together. The final sum is the answer to the summation problem.
