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Calculating individual terms of a sequence is essential to understanding its structure and behavior. In arithmetic sequences, each term is determined based on a specific formula. To calculate the terms of a sequence, such as in the exercise where the nth term is given by the formula a_n=n^2+2, we substitute consecutive integers for n to find the first few terms.

It’s important to perform the substitutions carefully. For the given sequence, we start with n=0 and find that the first term is 2, simply by plugging 0 into the given formula. Continuing with this process, n=1 yields 3, n=2 yields 6, n=3 yields 11, and n=4 yields 18. Each term is calculated in isolation, ensuring that the pattern, in this case, is followed properly.

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant difference, known as the common difference, to the previous term. The general formula to find the nth term of an arithmetic sequence is given by:
\[a_n = a_1 + (n - 1)d\]
where a_n is the nth term, a_1 is the first term, n is the position of the term in the sequence, and d is the common difference.

It's important to note that the sequence in the provided exercise, a_n = n^2 + 2, is not an arithmetic sequence, as the difference between successive terms is not constant. Instead, each term increases by an ever-larger amount because of the square function n^2. Recognizing whether a sequence is arithmetic or not is crucial as it determines which formula and methods to apply for calculations.

Mathematical induction is a powerful proof technique used in mathematics, particularly to prove that a statement is true for all natural numbers. The process involves two main steps:

Step 1 (Base Case): Verify that the statement holds for the initial value, usually n=0 or n=1.
Step 2 (Inductive Step): Assume the statement holds for an arbitrary natural number n=k. Then prove that the statement must also hold for n=k+1.

If both steps are successfully completed, it can be concluded that the statement holds for all natural numbers.

However, for the given example of finding the first five terms of a sequence, mathematical induction is not needed since we are not trying to prove a general statement. Instead, we directly calculate each term based on the formula provided. Mathematical induction would come into play if we wanted to prove a property that we claim holds for all terms of the sequence or series.