geometric progression

IPA/ˌdʒiːəˌmetrɪk prəˈɡreʃn/
IPA/ˌdʒiːəˌmetrɪk prəˈɡreʃn/

geometric progression — 名詞

1. a sequence created through repeated multiplication by the same value: starting f

1.名詞B2
釋義

等比數列

每項乘固定倍數的數列

a sequence created through repeated multiplication by the same value: starting from an initial number, you keep multiplying by a fixed quantity (the common ratio) to get the next number — for example, 2, 6, 18, 54, … or 1, 3, 9, 27, 81, …

例句

The teacher asked the class to find the next term in the geometric progression 5, 15, 45, 135.

老師要全班找出等比數列 5、15、45、135 的下一個數字。

finding the next term in a given GP

Li Wei saw that a geometric progression grows fast when the common ratio exceeds one.

Li Wei 發現當公比大於一的時候，等比數列會增加得非常快。

common ratio determines growth speed

同義詞
  • geometric sequence

    exactly the same meaning; 'sequence' and 'progression' are interchangeable in mathematics

  • geometric series

    often used for the sum of the terms of a geometric progression, though some textbooks use it interchangeably

反義詞
  • arithmetic progression

    a sequence where each term differs by a constant ADDITION or subtraction, not multiplication

用法筆記

A geometric progression has a constant multiplier (common ratio) between consecutive terms. This distinguishes it from an arithmetic progression, where the difference between terms is constant.

常見錯誤

❌The sequence 2, 4, 6, 8, 10 is a geometric progression.
✅The sequence 2, 4, 6, 8, 10 is an arithmetic progression.
💡In a geometric progression each term is multiplied by a fixed number; in an arithmetic progression a fixed number is added.
❌In a geometric progression the ratio is always between 0 and 1.
✅The ratio in a geometric progression can be any non-zero number, such as 3, 0.5, or −2.
💡The common ratio can be greater than 1, less than 1, positive, or negative, as long as it is not zero.