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Summation notation is a convenient way to express the addition of a sequence of numbers. This is particularly useful when dealing with long sequences where writing out each term individually would be impractical. For example, if we want to sum the first hundred positive integers, instead of writing them all out, we use summation notation as \( \sum_{i=1}^{100} i \).

In summation notation, the Greek letter sigma (\( \Sigma \) is used to denote the summation. The variable below this symbol indicates the starting index, and the number above signifies the endpoint of the summation. In our original exercise, \( \sum_{i=1}^{1} i(i+1) \), the summation runs from \( i=1 \) to \( i=1 \), meaning that it includes only one term where \( i \) is equal to 1.

Mathematical induction is a method of proof used to establish that a statement is true for all natural numbers. It is often used to prove the validity of statements involving sequences and series. The process of induction involves two main steps:

• Base Case: Show that the statement holds for the initial value, usually \( n=1 \).
• Inductive Step: Prove that if the statement holds for \( n=k \), where \( k \) is any natural number, then it also holds for \( n=k+1 \).

If both steps are correctly executed, it can be concluded that the statement is true for all natural numbers. Although the exercise used in our example does not directly involve induction, this method is a significant part of understanding how to generalize patterns found within series.

A sequence is an ordered list of numbers that often follows a particular rule or pattern, like the sequence of even numbers (2, 4, 6, 8, ...). A series is defined as the sum of the terms of a sequence. For instance, if we add the terms of the sequence of even numbers, we get a series. Sequences and series are fundamental concepts in discrete mathematics and can be finite or infinite.

In our original problem, we dealt with a finite series, specifically the sum of a single term given by the sequence \( a_i = i(i+1) \). Had the summation been over a larger range, say from 1 to \( n \), we would sum over the sequence to get the series \( \sum_{i=1}^{n} i(i+1) \). Understanding sequence and series is key to exploring more complex mathematical ideas like convergence, divergence, and can even connect to other areas of mathematics such as calculus with concepts such as infinite series.