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Geometric Sequences

Referencing Styles : APA | Pages : 1

Geometric sequence are depicted as a sequence of numbers which follows a particular pattern. In this pattern the next term of the series is found to be calculated after multiplying the prior number with a constant which is known as the common ratio that is r. The constant is termed as common ratio as the value which is derived by dividing two numbers within a series is always common.

The formula that depicts the geometric sequence is stated below:

an = an−1 ⋅ r or an = a1 ⋅ rn−1

The formula also states as well helps us to calculate a general term or the nth term. This nth term depicts the number of terms present within the series of a geometric sequence.

Let’s take an example that will tend to clarify the working of a geometric sequence:

We will find out the five terms of a geometric sequence in which a1 = 2 and r = 3.

In this case we will be using the provided formula and hence the answer to the given question is provided below:

a1 = 2

a2 = 2⋅3 = 6

a3 = 6⋅3 = 18

a4 = 18⋅3 = 54

a5 = 54⋅3 = 162

Hence the first five terms as per the given question are 2, 6, 18, 54 and 162.

Moreover the sum of a geometric sequence is found out with the help of the following formula:

Sn = a1 − a1 ⋅ rn/ (1−r) or Sn = a1 (1−rn) / (1−r)

The tendency to learn regarding this sequence is that the individuals sometimes faces the pattern of geometric sequences within the real life, and in this cases the individuals are in an urge of a formula which may help them to find a particular as well as significant number within the provided sequences. The proper definition of a geometric sequence is that it is a series of numbers in which the value of each previous number will be multiplied by a definite constant. The value of this constant may vary for each different series but the value remains constant with the overall individual series. Moreover, to check whether a provided sequence is geometric an individual simply checks that if the successive entries within the sequence possess the same ratio or not. The behaviour regarding a geometric sequence significantly depends on the values or numeric entities of common ratio. The conditions regarding the behaviour of a geometric sequence is stated as below:

  • If the provided common ratio is positive then the terms within the geometric sequence will be of the same sign as of the initial term.
  • If the provided common ratio is negative, then the terms will be alternatively between negative and positive.
  • If the provided common ratio is greater than 1, then there will be an exponential growth towards positive or negative infinity. However this depends on the sign of the initial term.
  • If the provided common ratio is 1, then the progression will be a constant one.
  • If the provided common ratio is between −1 and 1 but significantly not zero, then there will be an exponential tend of the value towards zero.
  • If the provided common ratio is −1,  then the progression will be in an alternating sequence
  • If the provided common ratio is less than −1, then there will be an exponential growth regarding the absolute value, in accordance to the alternating in the sign.

Now the main question that arises is that is this series quite relevant to the present modern world. It is a bit of surprise that it happens in real life as well as any individual might be involved within it by himself. Let us take a real life example. Suppose s teacher within a class gave an individual student the responsibility to convey a single message to all the other students that tomorrow it is mandatory for every student to wear a red scarf. The student who is provided with this responsibility texts a single message to five of his friends and then those five friends convey the same message to other students. Now observe the message chain that it started with a single individual and then it turned into 5 in the next stage. This can be depicted as an easy life example in regards to the geometric sequence. This text message is further conveyed to 25 people where each of the prior 5 students conveys the message to 5 more students. Thus in this case the geometric sequence is stated as 1, 5, 25, and 125 and so on.

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