Algebra Study Guide

A sequence is an ordered list of numbers, shapes or other mathematical objects that follow a specific pattern or rule. Each number in the sequence is called a term. To 'generate terms' of a sequence means to find the numbers that appear in the list, based on a given rule. There are two main ways a sequence rule might be given:

1. Term-to-term rule: This rule tells you how to get from one term to the next term. For example, 'add 3' or 'multiply by 2, then subtract 1'. To use a term-to-term rule, you always need to be given the first term of the sequence. Then, you apply the rule repeatedly to find subsequent terms.

1. Position-to-term rule (nth term rule): This rule tells you how to find any term in the sequence directly, based on its position. The position is usually represented by the letter 'n'. For example, an nth term rule might be '3n + 2'. To find the first term, you substitute n=1 into the rule. To find the second term, you substitute n=2, and so on. This method is often more efficient for finding terms far along in the sequence, as you don't need to calculate all the preceding terms.

It's important to understand which type of rule you've been given, as the method for generating terms is different for each. Pay close attention to whether the rule depends on the previous term (term-to-term) or the term's position (position-to-term).

A sequence is an ordered list of numbers, shapes, or other mathematical objects that follow a particular pattern or rule. Each number in the sequence is called a 'term'. Understanding how to generate terms of a sequence is a fundamental skill in mathematics. The rule for a sequence can be given in several ways, but a common one specifies how to find any term using its position number. This is often called the 'nth term' rule. For example, if the rule is 'add 3 to the previous term starting from 1', the sequence would be 1, 4, 7, 10, ... . If the rule is given algebraically, such as 'the nth term is 2n + 1', you find a term by substituting its position number (n) into the rule. For the 1st term, n=1, so 2(1) + 1 = 3. For the 2nd term, n=2, so 2(2) + 1 = 5. For the 3rd term, n=3, so 2(3) + 1 = 7. The sequence generated would be 3, 5, 7, ... . Generating terms of both arithmetic (constant difference between terms) and non-arithmetic sequences is important. Careful substitution and calculation are key to avoiding errors when generating terms.