“Just because we can’t find a solution it doesn’t mean that there isn’t one.” – Andrew Wiles
Definition
A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If a1 is the initial term of a geometric sequence and r is the common ratio, the sequence will be: {a1, a1r, a1r2, a1r3, …} [1]
The nth term of the geometric sequence is an = a · rn – 1.
Who
For anyone who was never introduced to arithmetic or geometric sequences in school (e.g., me) but had to deal with them on the ACT, GMAT and SAT. Sequences are essential not only in math but also computer science and other disciplines.
What
See articles on geometric sequences in the Additional Reading section below.
Why
We encounter geometric sequences in real life, and sometimes we need a formula to help us find a particular number in our sequence. We define our geometric sequence as a series of numbers, where each number is the previous number multiplied by a certain constant. This constant varies for each series but remains constant within one series.
Wondering how this kind of series relates to the real world? You might be surprised to learn that this kind of thing happens all the time, and you might even be involved in it yourself. Imagine that you wanted to spread the message that Friday is the new red scarf day, and everyone should wear a red scarf. First, you send your message to five of your friends. So, we started off with one person, and now we have five new people.
In the message, you ask your friends to send this message about the new red scarf day to five of their friends to help you spread the word. You also ask them to keep the ball rolling and ask their friends to each spread the word to five of their friends. Now we have 25 new people who know about the message. If each of these 25 people tells 5 of their friends, how many new people will know about the red scarf day? One hundred twenty-five people. So we have a geometric sequence of 1, 5, 25, 125… because each number is the previous multiplied by a constant, the common ratio. The common ratio in our case is five. Each of our numbers is multiplied by five to get to the next. [2]
See also Theoretical Knowledge Vs Practical Application.
How
I don’t show you how to to compute geometric sequences. Many of the References, Additional Reading, websites and videos will assist you with sequences.
As some professors say: “It is intuitively obvious to even the most casual observer.”
References
[1] Abramson, Jay. 2014. “Geometric Sequences”. openstax. https://opentextbc.ca/precalculusopenstax/chapter/geometric-sequences/.
[2] “How and Why to Use the General Term of a Geometric Sequence”. 2021. study.com. https://study.com/academy/lesson/how-and-why-to-use-the-general-term-of-a-geometric-sequence.html.
Additional Reading
⭐ “Geometric Progression – Definition, Formulas, Solved Examples”. 2020. GeeksForGeeks. https://www.geeksforgeeks.org/what-is-geometric-progression/?ref=lbp.
“Geometric Sequence”. 2021. math.net. https://www.math.net/geometric-sequence.
“Geometric Sequences”. 2021. nagwa.com. https://www.nagwa.com/en/videos/493151072621/.
“Geometric Sequences and Sums”. 2021. mathsisfun.com. https://www.mathsisfun.com/algebra/sequences-sums-geometric.html.
“Geometric Sequence Formulas”. 2023. CUEMATH. https://www.cuemath.com/geometric-sequence-formulas/.
The geometric sequence formulas include multiple formulas related to a geometric sequence. Before learning these formulas, let us recall what is a geometric sequence. It is a sequence of numbers in which the ratio of every two consecutive numbers is always a constant. For example, 2, 4, 8, 16, … is a geometric sequence as the common ratio of every two consecutive terms here is 2, i.e., common ratio = 4/2 = 8/4 = 16/8 = … = 2.
⭐ “Geometric Series – GeeksForGeeks”. 2021. GeeksForGeeks. https://www.geeksforgeeks.org/geometric-series/?ref=lbp.
Gregory Grant, and David K. “Why Is A Geometric Progression Called So?” 2015. Mathematics Stack Exchange. https://math.stackexchange.com/questions/1281856/why-is-a-geometric-progression-called-so?noredirect=1&lq=1.
“How To Solve Geometric Sequences”. 2020. effortlessmath.com. https://www.effortlessmath.com/math-topics/geometric-sequences/
“Number Sequence – Explanation & Examples”. 2021. The Story of Mathematics. https://www.storyofmathematics.com/number-sequences.
kuco. “Proofs Your Teacher Should Have Taught You”. 2022. Medium. https://nseverkar.medium.com/proofs-that-your-teacher-should-have-taught-you-1e3de9be2d87.
In high school, mathematics was done in what I now recognize as a very non-mathematical way. We were told to dogmatically memorize formulas and laws. Because of that, I struggled with many tests and got bad grades. That was until I realized what math truly was. It was not about memorizing stuff, it was about understanding it.
Suddenly everything became simple. Apart from the few definitions, I was able to derive pretty much everything and optimize my brain’s storage space. And later, at the university, this way of studying has turned out to be the right one.
This post is about some easy proofs that most teachers skip. We’re going to clear them up. They’re not difficult but can open a door to studying mathematics the way it was meant to be studied.
McGarry, Mike. 2012. “Sequences On The GMAT”. Magoosh Blog — GMAT® Exam. https://magoosh.com/gmat/sequences-on-the-gmat/.
“Make Your Kid’s Math Teacher Hate You”. 2022. Medium. https://medium.com/mathadam/make-your-kids-math-teacher-hate-you-a258d91dc2d7.
Montgomery, Courtney. 2021. “Sequences On ACT Math: Strategy Guide And Review”. blog.prepscholar.com. https://blog.prepscholar.com/sequences-on-act-math-strategy-guide-and-review.
“Sequences | Algebra 1 | Math | Khan Academy”. 2021. Khan Academy. https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences#x2f8bb11595b61c86:introduction-to-arithmetic-sequences.
“Sequences – Finding A Rule”. 2021. mathsisfun.com. https://www.mathsisfun.com/algebra/sequences-finding-rule.html.
Videos
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
The featured image on this page is from the nagwa website.
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