What is the fundamental difference between an arithmetic sequence and a geometric sequence?

An arithmetic sequence changes by adding or subtracting a constant value (common difference), whereas a geometric sequence changes by multiplying or dividing by a constant value (common ratio).

How does a quadratic sequence differ from a linear (arithmetic) sequence in terms of differences?

A linear sequence has a constant first difference. A quadratic sequence has changing first differences, but the second differences (the differences between the differences) are constant.

When comparing a Fibonacci sequence to an arithmetic sequence, what is the key difference in how the next term is found?

In an arithmetic sequence, the next term is found by adding a fixed constant to the previous term. In a Fibonacci sequence, the next term is found by adding the two previous terms together.

What error occurs if you only check the difference between the first two terms of a sequence?

You might misidentify the pattern because many different types of sequences (linear, quadratic, geometric) can share the same first two terms but diverge immediately afterward.

What is a common mistake when dealing with a decreasing arithmetic sequence?

Students often forget to treat the common difference as a negative number, which leads to incorrectly adding values instead of subtracting them when continuing the sequence.

Why is it a mistake to assume a sequence of fractions must have a single rule?

Fractions often consist of two independent sequences: one for the numerators and one for the denominators. Treating them as a single decimal value can obscure the simple integer patterns within.

Define the 'Common Difference' in the context of continuing a sequence.

The common difference is the constant value added to each term to get the next term in an arithmetic sequence, calculated as $d = u_{n+1} - u_n$.

What is a 'Common Ratio'?

The common ratio is the constant factor by which each term is multiplied to produce the next term in a geometric sequence, calculated as $r = \frac{u_{n+1}}{u_n}$.

What defines a 'Term' in a sequence?

A term is a single number within a sequence, identified by its value and its position ($n$) in the ordered list.

Why do we look for a 'second difference' if the first differences are not constant?

If the first differences change at a constant rate, it indicates the sequence is quadratic (related to $n^2$). The second difference reveals this underlying constant rate of change.

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Continuing Sequences

Summary

Continuing a sequence involves identifying the underlying mathematical rule that governs the progression of numbers and applying that rule to determine subsequent terms. This process relies on analyzing the relationships between consecutive terms through differences, ratios, or recognition of known power and recursive patterns.

1. Definition & Core Concepts

A sequence is an ordered list of numbers, where each individual number is referred to as a term ( $u_n$ ).
The position of a term is denoted by $n$ , where $n=1$ is the first term, $n=2$ is the second, and so on.
To continue a sequence, one must find the term-to-term rule, which describes how to get from one number to the next in the series.
A sequence is considered predictable if a consistent logical or arithmetic pattern can be established from the existing terms.

2. Underlying Principles of Pattern Recognition

Constant Differences: In many sequences, the change between terms is additive. If the difference is the same every time, the sequence grows linearly.
Successive Differences: If the first set of differences is not constant, calculating the 'difference of the differences' (second-order differences) can reveal quadratic patterns.
Proportional Growth: Some sequences grow by a constant multiplier rather than a constant sum, indicating a geometric relationship where each term is a fixed multiple of the previous one.
Recursive Logic: Certain sequences, such as Fibonacci-style patterns, determine the next term based on the sum or manipulation of two or more preceding terms.

A diagram showing a sequence of points on a graph with dashed lines representing the increasing differences between consecutive terms, illustrating a non-linear sequence.

3. Methods & Techniques for Continuation

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions