Difference Between Arithmetic and Geometric Sequence

What is Arithmetic Sequence?

Arithmetic sequence is defined as a sequence in which in the selected set of numbers there exists a constant difference between the preceding and succeeding number. The difference between each new phrase of numbers is fixed.

In arithmetic sequence the next number can be obtained by subtracting or adding the constant figure to the next or pervious number respectively. This can be illustrated as shown in the example below:

45,90,135,180,225

From the sequence above it is clear that the sequence has a constant difference of 45 which can be denoted by d. Therefore assuming that any number I the sequence is denoted by letter ā€˜nā€™ and kn for known number then it means that in order to find the previous or next number in the sequence we would use the following formulae;  kn-d =n for the example if the previous number is 225 and we wish to find the next then we will use 225+d=n which 225+45 then our number would be 270 hence the sequence would look like this: 45,90,135,180,225, 270

To find the previous number then the exact opposite would apply as follows.

45,90,135,180,—-? ,270

270-d=

270-45= 225 hence the missing number in the sequence would be 225.

What is Geometric Sequence?

A geometric sequence is a sequence in which the set of numbers have a common ration between them. In order to obtain the next number in the set sequence one is supposed to either divide of multiply using the common ratio.

For instance assuming the sequence 5,25,125,625,3125ā€¦ā€¦..

In the above sequence the common ration is 5 and all the subsequent number are obtained by multiplying the previous number by 5. Hence for instance, by multiplying the third number 125 by 5 we get 625 an so on.

In the same sequence  if we want to find the number 625 but have 3125 then we divide 3125 by 5 to get 625. This means that using the common ration of 5 we can either get the preceding or succeeding numbers in a selected sequence by either multiplying or dividing.

In a geometric sequence if the multiplier is greater than one then the succeeding numbers will always become bigger while the figures will become smaller if the multiplier is less that 1.

Difference Between Arithmetic and Geometric Sequence

  1. The main difference between arithmetic and geometric sequence is the mode in which the sequence is achieved. For arithmetic sequence there is a constant difference between the set in the sequence.
  2. In a geometric sequence the set of numbers is determined by the existing constant ratio.
  3. To find the succeeding number in an arithmetic sequence addition and subtraction is used to achieve the desired result.
  4. To find the succeeding number in an arithmetic sequence multiplication or division is used depending on the desired result.
  5. Finally in arithmetic sequence the variation is linear while in geometric sequence the variation is exponential.

Comparison Table Between Arithmetic and Geometric Sequence

Parameters of ComparisonArithmetic SequenceGeometric Sequence
DefinitionArithmetic sequence is a set of numbers in which the difference of succeeding values is determined by adding a constant figure.Geometric sequence is a set of numbers in which the succeeding number is determined by dividing or multiplying by a constant ratio.
IdentificationIdentified by common difference among given figures.Common ratio between the given figures.
VariationLinearExponential
Achieved bySubtraction or additionMultiplication or division.
Infinite sequenceThis can only be divergent.Can take either divergent or convergent approach.

References

  1. Rahmani-Andebili, Mehdi. “Problems: Arithmetic and Geometric Sequences and Series.” Precalculus. Springer, Cham, 2021. 101-105.
  2. Ross, Amanda. “A Brief Look at Arithmetic and Geometric Sequences.” Pedagogy and Content in Middle and High School Mathematics. Brill, 2017. 77-80.