27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
a. Find the next two terms of the sequence.
{-5, 5, -5, 5, ......}

Verified step by step guidance

1. Observe the given sequence: \(\{-5, 5, -5, 5, \ldots\}\). Notice the pattern of signs and values alternating between \(-5\) and \(5\).
2. Identify the rule governing the sequence. Since the terms alternate between \(-5\) and \(5\), the sequence can be described as \(a_n = (-1)^n \times 5\) or \(a_n = (-1)^{n+1} \times 5\), depending on the starting index.
3. To find the next two terms, determine the position of the last given term. The last term provided is the 4th term, which is \(5\).
4. Calculate the 5th term by applying the pattern: since the 4th term is \(5\), the 5th term will be \(-5\) (continuing the alternating sign pattern).
5. Similarly, calculate the 6th term, which will be \(5\), following the established alternating pattern.

Key Concepts

Sequences and Terms

A sequence is an ordered list of numbers defined by a specific rule. Each number in the sequence is called a term, denoted as aₙ, where n indicates its position. Understanding how terms relate to each other helps predict future terms.

Pattern Recognition in Sequences

Identifying the pattern or rule governing the sequence is essential. This involves observing how terms change from one to the next, such as alternating signs or repeating values, to determine the next terms accurately.

Alternating Sequences

An alternating sequence is one where the signs of the terms switch between positive and negative in a regular pattern. Recognizing this helps in predicting subsequent terms by applying the sign change rule consistently.