All of you must have been to movie theaters to watch movies with your friends or family members. While booking your tickets, have you ever noticed the way the seating arrangements are normally made at the movie theater? The number of seats in the previous row will always be lesser than the next row by a specific number.
This seating arrangement is normally in an arithmetic sequence. Thus, it can be said that a sequence that decreases or increases by a constant number is known as an arithmetic sequence. On the other hand, a geometric sequence is something completely different. Most of you have played with some sort of balls during your childhood days.
Whether you use a football or a basketball, you will notice that the height at which it bounces tends to decrease every time it hits the ground. This decrease in the bouncing height is in a geometric sequence.
Thus, it can be said that the geometric sequence is basically a sequence in which each term multiplies or divides by the same value from one specific term to the next one. The value by which a term divides or multiplies is known as the common ratio.
Arithmetic vs Geometric Sequence
The main difference between Arithmetic and Geometric Sequence is that while an arithmetic sequence has the difference between its two consecutive terms remains constant, a geometric sequence has the ratio between its two consecutive terms remains constant.
The difference between two consecutive terms in an arithmetic sequence is referred to as the common difference. On the other hand, the ratio of two consecutive terms in a geometric sequence is referred to as the common ratio.
Comparison Table Between Arithmetic and Geometric Sequence
|Parameter of Comparison||Arithmetic Sequence||Geometric Sequence|
|Definition||It is a list of numbers, in which every new term alters from another preceding term by a definite quantity.||It is a sequence of numbers in which each new term is calculated by multiplying by a non-zero and fixed number.|
|Calculated By||Addition or Subtraction||Multiplication or Division|
|Identified By||A constant difference between 2 successive terms.||A common ratio between 2 successive terms.|
|Form||Linear Form||Exponential Form|
What is Arithmetic Sequence?
When you talk about arithmetic sequence or arithmetic progression, it basically refers to a sequence of different numbers in which the difference between 2 consecutive numbers is always constant.
In this type of sequence, difference means the first term subtracted from the second term. If you consider a sequence such as 1, 4, 7, 10, 13…it is an arithmetic sequence in which the constant difference if 3.
Just like anything else in mathematics, an arithmetic sequence also has a formula. The formula used to find an arithmetic sequence is a, a+d, a+2d, a+3d, and so on. In this formula, “a” is the first term and “d” is the common difference between 2 consecutive terms.
It is important for you to know that the behavior of an arithmetic sequence depends a lot on the common difference. If the common difference or the “d” in the formula is positive, then the terms will grow in a positive manner. However, if the common difference is negative, the terms will grow in a negative manner.
What is a Geometric Sequence?
The geometric sequence or geometric progression in mathematics happens to be a sequence of different numbers in which each new term after the previous is calculated by simply multiplying the previous term with a common ratio. This common ratio is a fixed and non-zero number. As an example, the sequence 3, 6, 12, 24, and so on is a geometric sequence with the common ratio being 2.
A geometric sequence also has a formula of its own. The normal form of a geometric sequence is in the form of a, ar, ar², ar³, ar4 and so on.
When you need to find the n-th term in any geometric sequence, the formula to use is an = arn-1, where the common ratio “r” and the initial value “a” will be given. There are certain factors you should remember when it comes to a geometric sequence. If the common ratio is positive, the terms will also be positive.
However, if the common ratio is negative, the terms will be alternate between negative and positive. If the common ratio is greater than 1, the growth will be in an exponential form towards positive or even negative infinity. If the common ratio is 1, then the progression will be a constant sequence.
Main Differences Between Arithmetic and Geometric Sequence
- An arithmetic sequence is a sequence of numbers that is calculated by subtracting or adding a fixed term to/from the previous term. However, a geometric sequence is a sequence of numbers where each new number is calculated by multiplying the previous number by a fixed and non-zero number.
- The difference between two consecutive terms in an arithmetic sequence is known as the common difference that is represented by “d”, and the number by which terms multiple or divide in a geometric sequence is known as the common ratio represented by “r”.
- When it comes to an arithmetic sequence, the variation is in a linear form. On the other hand, when it comes to a geometric sequence, the variation is in an exponential form.
- In an arithmetic sequence, the numbers may either progress in a positive or negative manner depending upon the common difference. Whereas, in a geometric sequence there is no such rule as the numbers may progress alternatively in a positive and negative manner in the same sequence.
Frequently Asked Questions (FAQ) About Arithmetic and Geometric Sequence
Why is it called a Geometric Sequence?
It’s called a geometric sequence because the numbers go from one number to another by diving or multiplying by a similar value.
The number divided or multiplied at every stage of the series called the common ratio. A geometric series is a set of figures that follow a unique rule of a pattern.
Can an Arithmetic Sequence also be a Geometric?
In math, an arithmetic series is defined as the sequence where the variance between consecutive numbers called the common difference is constant.
On the other hand, the geometric series is where the ratio between successive numbers, known as a common ratio, is constant. So, that means a sequence can’t be both geometric and arithmetic.
What is the infinite Geometric Series formula?
The infinite geometric sequence is defined as a totality of an infinite geometric sequence. The sequence doesn’t have the last figure. This type of an infinite sequence include a1+a1r+a1r2 +a1r3+…. In this case, a1 refers to the first figure while r refers to the common ratio.
You will calculate the total sum of a finite geometric sequence. In the case of the infinite geometric sequence, once the common ratio is above one, the terms in the series will increase, and when you add larger numbers, getting a final answer will be impossible. The only answer would be infinity.
Let’s say the r (common ratio) lies between -1 and 1/. You can get the sum of an infinite geometric sequence. That’s, the sum exists for r <1.
The sum of infinite geometric series that has -1<r<1 is calculated by:
What is A in an arithmetic sequence?
An arithmetic sequence refers to the series of terms such that a difference between two successive participants of the series is a constant term whereby a in the arithmetic sequence is the first term.
How do you find the nth term of an arithmetic sequence?
The terms in an arithmetic series are known to increase by the common difference (d). For instance, 2, 4, 6, 8, 10 is an arithmetic progression and d=2.
The formula to get the nth term of this arithmetic sequence is 2n+1. Typically, the nth term of an arithmetic sequence with a1st term and a common difference is a+ (n-1) d.
With the help of this detailed discussion about the differences between an arithmetic sequence and a geometric sequence, you should be clear about it by now. If you think that these 2 sequences do not have any real-life uses, then you should think again. Both have their individual uses and importance in different day to day lives.
Arithmetic sequences are used in various financial sectors and can prove to be rather useful when it comes to calculating your savings and personal financial increments. However, a geometric sequence also has its fair share of uses. It is used to calculate interest rates provided by different financial institutions and also to calculate the population growth of a country.
It is often seen that students get confused when it comes to deciding whether a given sequence is an arithmetic sequence or a geometric sequence. Although calculating an arithmetic sequence is pretty simple, the main challenge lies in calculating a geometric sequence.