91Ó°ÊÓ

An arithmetic sequence is a list of numbers where each term is followed by a constant value called the common difference. It's important to know how to find the terms of such a sequence as this is fundamental to solving many problems in mathematics.

To determine the terms of an arithmetic sequence, you often start with a given formula or equation. In this case, we have the expression \(a_n = 5 + 3n\). This equation allows us to find each term in the sequence by simply plugging in the integer values of \(n\).

For \(n = 1\), substitute \(n\) into the formula: \(a_1 = 5 + 3 imes 1 = 8\).

Continue this for \(n = 2, 3, 4, 5\), resulting in terms \(11, 14, 17,\) and \(20\) respectively. These are the first five terms of the sequence, and it's essential to calculate these accurately to verify if the sequence is arithmetic.

Once you have several terms of a sequence, the next task is to check if the sequence is arithmetic. For an arithmetic sequence, the difference between consecutive terms should always be the same. This difference is called the "common difference".

To find the common difference, you subtract any term \(a_n\) from the subsequent term \(a_{n+1}\). After finding the first few terms, look at the differences between them:

• \(11 - 8 = 3\)
• \(14 - 11 = 3\)
• \(17 - 14 = 3\)
• \(20 - 17 = 3\)

Since all these differences are equal to 3, we can confidently say the sequence is arithmetic, and the common difference \(d\) is \(3\). This regularity is what guarantees the sequence's arithmetic nature.

The substitution method is widely used in sequences to extract information from formulas or equations. It's a simple and effective way to generate terms of a sequence by substituting numerical values into a given expression.

In the context of arithmetic sequences, such as the one given by the expression \(a_n = 5 + 3n\), the substitution involves replacing \(n\) with successive integer values starting from 1. This direct substitution helps in quickly determining each term of the sequence without manually calculating each step separately.

For example, when you set \(n = 2\), substituting this into the formula gives \(a_2 = 5 + 3 imes 2 = 11\).

This systematic substitution for each subsequent \(n\) helps verify if the sequence's characteristics, such as its arithmetic nature, hold true, and it assists in identifying patterns or calculating missing terms.