91Ó°ÊÓ

Sigma notation, represented by the symbol \(\backslash sum\), is a convenient way to represent the sum of a sequence of terms. The notation can often look a bit intimidating at first, but once you break it down, it becomes much easier to understand.

• Upper and Lower Limits: The numbers written at the top and bottom of the sigma symbol are the limits of summation. In this case, \(i\) ranges from 1 to 3.
• Expression: The expression to the right of the sigma symbol is what’s being summed. In this case, it is \(i^{2} + 2\).

To evaluate the series using sigma notation, you simply replace \(i\) with each integer value from the lower limit to the upper limit, calculate the result for each, and then add all these results together.
This makes sigma notation very powerful for compactly writing sums involving many terms.

A quadratic expression is a polynomial of degree 2, which means it includes a term with the variable squared (\footnote\(x^{2}\)). In the given exercise, the quadratic expression is \(i^{2} + 2\). Breaking it down:

• \(i^{2}\) is the squared term or the quadratic term.
• \(2\) is a constant term.

Quadratic expressions are essential in many areas of mathematics and often show up in problems involving parabolic motion, area calculations, and optimization problems.
In our example, we evaluate the quadratic expression by substituting the given values of \(i\) one by one. Calculating separately for \(i=1, 2, 3\):
\[i=1 : (1^{2} + 2 = 3)\]
\[i=2 : (2^{2} + 2 = 6)\]
\[i=3 : (3^{2} + 2 = 11)\].
Then, we sum those calculated values: \(3 + 6 + 11 = 20\).
It's helpful to break down each step to simplify the problem.

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. In general, the formula for the sum of the first \(n\) terms of an arithmetic series is:
\[ S_n = \frac{n}{2} \times (a + l) \]
where \(a\) is the first term and \(l\) is the last term.
Although the given exercise involves a quadratic expression, not strictly an arithmetic series, understanding how sequences are summed up is critical. Here, the sequence of values generated by evaluating \(i^{2}+2\) at \(i = 1, 2, 3\) has unique values rather than a common difference.
Summing sequences, whether arithmetic or not, emphasizes adding all evaluated terms correctly.
In our exercise:
\[3 (result for i=1) \]
\[+ 6 (result for i=2) \]
\[+ 11 (result for i=3) \]
\[ = 20 \].
Thus, understanding arithmetic series helps in recognizing patterns and systematic summing, even in more complex series.