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Sequence formulas are mathematical expressions used to describe patterns in sequences of numbers. These patterns help predict subsequent terms without the need to list them all. The general form of a sequence can vary, but an arithmetic sequence, like the one in this exercise, is often defined using a formula such as \( a_n = 3n + 1 \). This particular formula indicates that each term of the sequence can be found by multiplying the term number \( n \) by 3 and then adding 1.

Understanding sequence formulas is crucial because it allows you to quickly determine any term in the sequence without having to sequentially calculate each preceding term. This capability is especially powerful when dealing with large indices that require extensive computation.

When working with sequence formulas, always double-check that you clearly understand the structure & pattern they represent, as this will make calculations much easier and less error-prone.

In arithmetic sequences, calculating individual terms involves substituting values into the sequence formula. For the sequence \( a_n = 3n + 1 \), calculating terms one by one is straightforward:

• To find the first term \( a_1 \): substitute \( n=1 \) into the formula, resulting in \( a_1 = 3(1) + 1 = 4 \).
• To get the second term \( a_2 \): replace \( n=2 \) in the formula to get \( a_2 = 3(2) + 1 = 7 \).


• For the third term \( a_3 \): substitute \( n=3 \) yielding \( a_3 = 3(3) + 1 = 10 \).
• Finally, the fourth term \( a_4 \): use \( n=4 \) in the formula to find \( a_4 = 3(4) + 1 = 13 \).

These calculations demonstrate how straightforward it is to derive terms from a given sequence formula. Familiarity with this process ensures accuracy and efficiency when determining individual terms or when verifying patterns presented in problems.

Mathematical substitution plays a critical role in solving exercises involving sequences. It refers to replacing variables in an equation or formula with actual numbers to calculate specific terms or solve expressions.

In the context of sequences such as the one in the provided exercise, substitution is used to replace \( n \) with specific integers to find the corresponding terms. With the formula \( a_n = 3n + 1 \), performing substitution involves following these steps:

• Select the term number \( n \) you want to calculate.
• Substitute this value into the sequence formula in place of \( n \).
• Solve the resulting expression to find the term.

Despite being straightforward, substitution is a key mathematical procedure used widely across different areas of math and science. Mastery of substitution not only helps with sequence problems but also aids in solving equations and modeling real-world situations efficiently and accurately.