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An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant, known as the common difference, to the previous term. For example, in the sequence 2, 4, 6, 8, the common difference is 2, because each term is 2 more than the previous term.

The formula to find the nth term of an arithmetic sequence is given by: \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

To determine whether a sequence is arithmetic, you can subtract consecutive terms to see if the difference remains constant. However, the sequence given in the exercise \(a_n = \frac{{(-1)^n}}{{n^2}}\) is not arithmetic because the difference between terms is not constant.

In contrast to arithmetic sequences, geometric sequences are those where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Take the sequence 3, 9, 27, 81 for instance; the common ratio here is 3 since every term is three times the previous term.

The formula for the nth term of a geometric sequence is: \(a_n = a_1 \cdot r^{(n-1)}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) represents the term number. To identify if a sequence is geometric, you can divide consecutive terms to check if the ratio is consistent. The sequence from our exercise \(a_n = \frac{{(-1)^n}}{{n^2}}\) does not fit this pattern either, as the ratio between terms changes, making it neither arithmetic nor geometric.

A sequence term formula allows us to calculate the nth term of a sequence without listing all the preceding terms. It serves as a shortcut to directly find any term in the sequence. The sequence from the exercise has the term formula \(a_n = \frac{{(-1)^n}}{{n^2}}\).

Applying the formula, you simply substitute the term's position (n) into the formula to find its value. This direct approach saves time and makes finding terms of sequences with large indices feasible. It's a powerful tool for understanding and working with sequences in mathematics.

Mathematical induction is a method used to prove a wide range of mathematical statements, particularly those formulated in terms of n. It works in a 'domino effect' manner: if you prove the statement for the first case (usually n=1), and then show that if any one case is true, the next case (n+1) must also be true, you've proven the statement for all natural numbers.

The process consists of two main steps: the base case and the inductive step. Success in applying mathematical induction involves carefully defining these steps and showing the logical progression from one case to the next. Although not used for finding terms in sequences, it's instrumental in proving properties about them. For example, you could use induction to prove certain characteristics of the sequence in the exercise if needed.