How does sequence and progression differ?
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Evans Gyedublay, Undergraduate Medicine and Healthcare & Science, Université Mohammed Premier Oujda (2023)
Sequences are a set of numbers, which are arranged according to any specific rule. There is no exception for any type of numbers, any type of rules according to which they are arranged. The set of numbers should have a definite, logical rule according to which they are arranged. It need not be a mathematical formula, but it should be logical. Such a set of numbers are called a sequence of numbers. sequenceは任意のルールに従って並べられた数字の集合です。どの種類の数字もその数字が並べられられたルールにも例外はありません。数字の集合はそれらが並べられる明確な、論理的なルールを持っているはずです。それは数学的な式である必要はありませんが、論理的であるべきです。そのような数字の集合が数字のsequenceと呼ばれます。
For example, the following is a sequence of numbers, because they are arranged according to a definite rule:
{2, 4, 6, 8, 10, 12} Rule: nth term = 2n
{3, 5, 7, 9, 11, 13} Rule: nth term = 2n + 1
The following is also a sequence of numbers, as they too have a logical rule:
Progressions
Progressions are yet another type of number sets which are arranged according to some definite rule. The difference between a progression and a sequence is that a progression has a specific formula to calculate its nth term, whereas a sequence can be based on a logical rule like 'a group of prime numbers', which does not have a formula associated with it. progressionはさらにもう一つの明確なルールに従って並び替えられた数字の集合です。prgoressionとsequenceの間の違いは、prgoressionはそのn番目の項を計算するある式を持っているのに対して、sequenceは「素数のグループ」のような、それと結びつく式を持たない論理的なルールに基づいていることがあります。 Now you may be wondering that a set of prime numbers should a progression because we can predict its nth term, but a progression needs a specifically stated formula, and, it is to be noted that prime numbers cannot be predicted with the help of any formula; Till date, the formula for the nth prime number has not be found. This means that we can only calculate the nth prime number with the method of selecting each successive number and checking whether it is prime or not. 今あなたは何番目の項かを予想することができるから素数の集合はprogressionであるはずだと不思議に思っているかもしれませんが、progressionは明確に定められた式が必要で、さらに素数はいかなる式の助けを持ってしても予想することができないことを覚えておくべきです。現在のところ、素数の式は見つかっていません。これは私達がそれぞれの連続する数を選択してそれが素数かどうかを確認する方法でしかn番目の素数だと計算することしかできないことを意味します。
Therefore {2, 4, 6, 8, 10, 12} represents a progression where the nth term is given by '2n', and, on the other hand, {2, 3, 5, 7, 11, 13, 17} represents a sequence that is not a progression, because although it is based on a definite logical rule (prime numbers), but there is no formula to calculate its nth term. You may also be wondering why there is no formula for prime numbers till today. The answer is that no one has been able to find the formula for prime numbers. Maybe you can? :) あなたはなんで今日まで素数の式がないのかとも不思議に思っているかもしれません。その答えは誰も素数の式を見つけることができなかったということです。あなたはできるかもしれませんね。
Coming back to our topic, we can conclude that a sequence is a set of numbers arranged according to some rule, a series is a sequence separated by + signs, and a progression is a sequence that can be stated by a specific mathematical formula like nth term = (2 + n).
私達の議題に戻ると、sequenceはいくつかのルールにしたがって並べられた数字の集合、seriesは+記号で分けられたsequence、そしてprogressionはある数学的な式によって定められるものであると結論付けることができます。
We also observe that all series are based on specific sequences, and that although all progressions are sequences, all sequences are not progressions(for example, prime numbers 私達はすべてのseriesは特定のsequenceに基づいているが、すべてのprogressionはsequenceであり、すべてのsequneceはprogressionではないということも同様に気づきます。