The first of these is the one we have already seen in our geometric series example.
There exist two distinct ways in which you can mathematically represent a geometric sequence with just one formula: the explicit formula for a geometric sequence and the recursive formula for a geometric sequence. Now that we understand what is a geometric sequence, we can dive deeper into this formula and explore ways of conveying the same information in fewer words and with greater precision. These other ways are the so-called explicit and recursive formula for geometric sequences. However, there are more mathematical ways to provide the same information. This allows you to calculate any other number in the sequence for our example, we would write the series as: 1, 2, 4, 8. Where n is the position of said term in the sequence.Īs you can see, the ratio of any two consecutive terms of the sequence - defined just like in our ratio calculator - is constant and equal to the common ratio.Ī common way to write a geometric progression is to explicitly write down the first terms. The n-th term of the progression would then be: aₙ = 1 * 2ⁿ⁻¹, In this case, the first term will be a₁ = 1 by definition, the second term would be a₂ = a₁ * 2 = 2, the third term would then be a₃ = a₂ * 2 = 4 etc. To make things simple, we will take the initial term to be 1 and the ratio will be set to 2.
Now, let's construct a simple geometric sequence using concrete values for these two defining parameters. We will see later how these two numbers are at the basis of the geometric sequence definition and depending on how they are used, one can obtain the explicit formula for a geometric sequence or the equivalent recursive formula for the geometric sequence. This common ratio is one of the defining features of a given sequence, together with the initial term of a sequence. A geometric sequence is a collection of specific numbers that are related by the common ratio we have mentioned before. Now let's see what is a geometric sequence in layperson terms. But if we consider only the numbers 6, 12, 24 the GCF would be 6 and the LCM would be 24. For example, in the sequence 3, 6, 12, 24, 48 the GCF is 3 and the LCM would be 48. Conversely, the LCM is just the biggest of the numbers in the sequence. This means that the GCF is simply the smallest number in the sequence. Indeed, what it is related to is the greatest common factor (GFC) and lowest common multiplier (LCM) since all the numbers share a GCF or a LCM if the first number is an integer. First of all, we need to understand that even though the geometric progression is made up by constantly multiplying numbers by a factor, this is not related to the factorial. We also include a couple of geometric sequence examples.īefore we dissect the definition properly, it's important to clarify a few things to avoid confusion. If you are struggling to understand what a geometric sequences is, don't fret! We will explain what this means in more simple terms later on, and take a look at the recursive and explicit formula for a geometric sequence. The geometric sequence definition is that a collection of numbers, in which all but the first one, are obtained by multiplying the previous one by a fixed, non-zero number called the common ratio.
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