Evaluate the Summation sum from x=1 to 3 of x^2-1
The given problem asks to calculate the sum of a sequence of values generated by the expression x^2-1, where x takes on integer values starting at 1 and ending at 3. The summation operator indicates that you need to evaluate the expression for each individual value of x within the range (which are 1, 2, and 3), and then add all these results together to get the final sum.
$\sum_{x = 1}^{3} x^{2} - 1$
Answer
Solution:
Step 1:
Write out the terms of the summation for each integer value of $x$ from $1$ to $3$.
$$(1^2 - 1) + (2^2 - 1) + (3^2 - 1)$$
Step 2:
Perform the calculations step by step.
Step 2.1:
Calculate $1^2$.
$$1^2 - 1 + 2^2 - 1 + 3^2 - 1$$
Step 2.2:
Subtract $1$ from the result of $1^2$.
$$0 + 2^2 - 1 + 3^2 - 1$$
Step 2.3:
Calculate $2^2$.
$$0 + 4 - 1 + 3^2 - 1$$
Step 2.4:
Subtract $1$ from the result of $2^2$.
$$0 + 3 + 3^2 - 1$$
Step 2.5:
Combine the results obtained so far.
$$3 + 3^2 - 1$$
Step 2.6:
Calculate $3^2$.
$$3 + 9 - 1$$
Step 2.7:
Subtract $1$ from the result of $3^2$.
$$3 + 8$$
Step 2.8:
Add the final terms together to obtain the sum.
$$11$$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a process of adding up a sequence of numbers generated by running through a series of values for a variable, in this case, $x$. The summation is defined by the function $x^2 - 1$, and we are summing over the integer values of $x$ from $1$ to $3$.
To solve this problem, we:
Expand the Series: We substitute each integer value of $x$ into the function $x^2 - 1$ and write out the terms of the series.
Simplify the Series: We then simplify each term by performing the operations indicated:
Squaring the number: We raise each integer value of $x$ to the power of $2$.
Subtracting one: We subtract $1$ from the squared value.
Adding the terms: We add the simplified terms together to find the sum.
Perform Arithmetic Operations: We perform the basic arithmetic operations of addition and subtraction as required to simplify each term and ultimately find the sum.
In this problem, the use of the summation symbol $\sum$ indicates that we are to sum the values of the function $x^2 - 1$ for each integer $x$ from $1$ to $3$. The solution involves basic arithmetic and the properties of exponents. The final result is a single number, which is the sum of the series.
