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In any arithmetic sequence, the formula for the general term is typically written as \(a_n = a_0 + nd\), where \(a_0\) is the first term and \(d\) is the common difference between the terms. This common difference is reflected in the sequence's formula as the coefficient of \(n\). For the sequence given in the exercise, the formula is \(a_n = 3 + 5n\). Here, the number 5, which is multiplied by \(n\), is the coefficient of \(n\). This number indicates how much each term increases as \(n\) increases by 1.

Understanding the role of the coefficient is key to grasping arithmetic sequences. The coefficient of \(n\) determines the step size of the sequence. In this sequence, it tells us that each step, or movement from one term to the next, involves adding 5 to the previous term. Thus, if you know the coefficient of \(n\), you can quickly identify how the terms of the sequence progress.

The zeroth term, often represented as \(a_0\), is the starting point of an arithmetic sequence. Typically, it's the value of the sequence when \(n=0\). Establishing the zeroth term provides a way to calculate further terms in the sequence.

In the example provided, substituting \(n=0\) into the sequence formula \(a_n=3+5n\) yields \(a_0=3+5(0)=3\). Therefore, the zeroth term \(a_0\) is 3. This value acts as the base from which the sequence builds; every subsequent term adds multiples of the common difference—illustrated by the coefficient of \(n\)—to this zeroth term.

By understanding the significance of \(a_0\), you can not only derive the specific terms of the sequence but also comprehend the initial point from where the sequence begins and how it evolves over each step.

Substitution in mathematics involves replacing variables with numbers or other expressions to simplify a problem. In the context of finding specific terms in a sequence, substitution is used to evaluate the general formula for different values of \(n\).

For instance, to find the zeroth, first, and second terms in the sequence \(a_n=3+5n\), we use substitution:

• Set \(n=0\) to find \(a_0\): \(a_0 = 3 + 5 \, (0) = 3\).
• Set \(n=1\) to find \(a_1\): \(a_1 = 3 + 5 \, (1) = 8\).
• Set \(n=2\) to find \(a_2\): \(a_2 = 3 + 5 \, (2) = 13\).

This technique allows you to identify and calculate the terms of a sequence accurately by systematically inserting different values of \(n\) into the sequence formula and solving the resulting expression.