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An arithmetic sequence is a list of numbers with a specific pattern where the difference between consecutive terms remains constant. This difference is known as the "common difference". In an arithmetic sequence, each term after the first is created by adding the common difference to the previous term.
For example, in the sequence 3, 6, 9, 12, ..., each number increases by 3. So, 3 is the common difference. To determine if a sequence is arithmetic, check this difference between each pair of consecutive terms. If it's consistent throughout, it's arithmetic.

• Constant addition from one term to the next.
• The common difference can be positive, negative, or zero.
• If the common difference is zero, all numbers in the sequence are the same.

Understanding arithmetic sequences is crucial since they often appear in various real-world scenarios, such as calculating loan payments or analyzing supply chains.

The common ratio in a geometric sequence is what sets it apart from an arithmetic one. It's a factor by which we multiply each term to get to the next one. To find it, simply divide a term by its preceding term.
Consider the sequence 2, 6, 18, 54, .... If you divide 6 by 2, you get 3. Do the same with 18 by 6, and you also get 3 again with 54 divided by 18. So, 3 is the common ratio here. This constancy confirms that the sequence is geometric.

• Multiplication by the same factor to advance through the sequence.
• The common ratio can be any non-zero number, including fractions or negative numbers.
• If the common ratio is greater than 1, the terms increase; if it's between 0 and 1, the terms decrease.

Knowing the common ratio is essential, especially when modeling exponential growth or decay scenarios like population growth or radioactive decay.

It's important to distinguish between arithmetic and geometric sequences since they differ fundamentally.
Arithmetic sequences rely on addition or subtraction, having a constant difference, while geometric sequences depend on multiplication or division, maintaining a constant ratio.
To decide which kind of sequence you're dealing with, first look for a pattern. Try subtracting consecutive terms to check for an arithmetic sequence. If the results are inconsistent, try dividing terms to see if a common ratio exists, indicating a geometric sequence.

• Arithmetic sequences: consistent addition/subtraction.
• Geometric sequences: consistent multiplication/division.

These differences make understanding both types of sequences essential in tasks like financial forecasting or sequence algorithms, as each applies to different mathematical and real-life contexts.