geometric series

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Word: Geometric Series

Definition: A geometric series is a mathematical term that describes a sequence of numbers where each number after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio." When we write this sequence as a sum, we call it a geometric series.

Usage Instructions:

• You typically use "geometric series" in math or when discussing sequences and series in algebra.
• It often appears in topics related to calculus, finance, and computer science.

Example:

Consider the geometric series where the first number is 2 and the common ratio is 3. The series would look like this: - 2, 6, 18, 54, ...

Advanced Usage:

In a more advanced context, you might encounter formulas to find the sum of a geometric series. For example, if the first term is "a" and the common ratio is "r," the sum of the first "n" terms can be calculated using the formula: [ S_n = a \frac{(1 - r^n)}{(1 - r)} ] (if ( r \neq 1 ))

Word Variants:

• Geometric Progression: This is often used interchangeably with geometric series but refers to the sequence itself rather than the sum.
• Common Ratio: The number you multiply by to get from one term to the next in the geometric progression.

Different Meanings:

• Geometric: This word relates to shapes and sizes in mathematics and can describe anything that is formed by lines and angles.
• Series: In general, this word can refer to any sequence or set of related items.

Synonyms:

• Geometric Sequence: Refers to the same concept of numbers in a sequence multiplied by a common ratio, but focuses on the order rather than the sum.

Idioms and Phrasal Verbs:

There aren't specific idioms or phrasal verbs that directly relate to "geometric series," as it is mostly a mathematical term. However, you might encounter phrases like: - "The series converges," which means that as you keep adding more terms, the total approaches a specific number. - "Diverges," meaning the series does not settle down to a single value.

Summary:

A geometric series is an important concept in mathematics that helps us understand how numbers can grow or shrink in a predictable pattern based on multiplication. It has applications in various fields, including finance (like calculating compound interest) and computer science (like algorithms).

Academic

Noun

1. a geometric progression written as a sum