Introduction
In the context of economic and financial growth calculations, it is important to know about the geometric sequence and the arithmetic means. These terms are used extensively in areas such as stock markets, increments, population, and other important areas.
Arithmetic deals with mathematical operations that deal with numerical systems and their related operations. It is used to obtain a single, definitive value. "Arithmetic" is derived from the Greek word "arithmos," meaning "numbers." This field of mathematics focuses on the study and properties of common operations, such as subtraction, multiplication, division, addition, and subtraction.
A sequence is a group of items in a particular order (typically numbers). The most common types of mathematical sequences are arithmetic and geometric. Arithmetic sequences have a constant difference between each pair of terms. A geometric sequence on the other side has a fixed ratio between each pair.
Arithmetic Mean vs. Geometric Sequence
Geometric Sequence and Arithmetic Mean have one thing in common. Arithmetic Mean is used to calculate the average of a collection of numbers, while geometric sequence simply collects numbers with constant ratios.
The Arithmetic average, or simply the average, is the sum of the numbers divided by the number of numbers. While the Geometric sequence refers to the set of terms obtained by multiplying or dividing the constant term,
A sequence is a structured collection of terms that follow a consistent pattern. The 'arithmetic average' is the average of the sequence of numbers. These mathematical terms are used often to determine the methodical organization of terms.
The arithmetic average is the sum of numbers in a sequence. If two terms are separated by a constant number, the ratio is determined using the geometric sequence called a common ratio.
Difference Between Arithmetic Mean and Geometric Sequence in Tabular Form
Parameters for Comparison | Arithmetic means | Geometric sequence |
Definition | The average of all numbers in a sequence is called the arithmetic median. | The geometric sequence refers to a collection of terms where the difference between two consecutive terms is constant. |
Decided by | This can be done by multiplying the sum of all the numbers by the total number of numbers. | You can determine it by multiplying or dividing a constant with the preceding term. |
Form | This is the average of all the collections. | This sequence is often expressed in an exponential form. |
Common formula | A = (a1+ a2+.. + an)/n (where A is the 1st Digit and n is how many digits it takes to find the mean A using this formula). |
tn = 1 r(n - 1)
(where r is a common ratio and where tn is nth term, while t1 is the first term). |
Uses | In the experimental and observational studies, the average or the arithmetic means is used to give a quick idea of the large sample size. The central tendency of the data becomes then the mean. | In various economic and financial sectors, a geometric sequence is used to calculate growth rates, savings, and costs. |
What does Arithmetic Mean?
The arithmetic average is the sum of all terms that can or cannot be separated by the common differentiation. The sum of all the terms is divided by the total number of numbers to find the mean. Because'mean' always refers to the central tendency of data, it is easy to find the average or arithmetic means for reducing large sample sizes.
The mean is the sum of all observations and observational research. It can be written as:
Arithmetic Mean = sum of all observations/total number
If the data is in a sequence, the formula can calculate the average sequence.
A= (a1 + a2+ .. + an)/n
"A" is the average or the arithmetic means, "a" is the first term, and "n" is the total number of terms in the collection.
We need to find the arithmetic means of the sequences 2, 4, 6, 8, and 10, for example.
This is easily accomplished by using the formula (2+4+6+8+10)/5= 6.
When observed, the arithmetic average can be used in everyday life. The average is important in anthropology, history and statistics to calculate per capita income. Because it only approximates the value, the arithmetic average has some limitations. The average cannot be used in financial data, where each term is important.
Arithmetic Mean Properties
These are some important properties of the arithmetic means:
- The sum of all items that are not within their arithmetic means is always zero. (x - X) = 0.
- The sum of squared deviations from Arithmetic Mean items (A.M) is minimal, which is lower than the sums of squared deviations from other values.
- If every item in an arithmetic series can be substituted by its mean, the sum will equal the sum of those specific items.
Arithmetic Mean for Ungrouped Data
Ungrouped data can be easily found in the arithmetic means by adding all of the values to a data set, and then dividing it by a number.
Mean, x = Sum all values/Number
Example: Find the arithmetic means of 4, 8, 12, 16, and 20.
Solution: Given, that 4, 8, 12, 16 and 20 are the values.
Sum of all values = 4+8+12+16+20= 60
Values = 5.
Average = 60/5 = 12
Merits of Arithmetic Mean
- It is easy to comprehend and calculate the arithmetic mean.
- It is affected by the individual items' values.
- A.M. is rigidly defined.
- It can be further algebraic treated.
- It is not based on position in the series.
Demerits of Arithmetic Measure
- Extreme items, such as extremely small or very large items, can alter this.
- Inspection is rarely able to identify it.
- Sometimes, A.M. may not be the original item. The average hospital admission rate is 10.7 patients per day.
- In extremely asymmetrical distributions, the arithmetic median is not appropriate.
Representative Data Values
Representative value is a common concept in everyday life. If you inquire about the mileage of your car, you will be asking for the representative value of how far you have travelled and the fuel consumption. It doesn't necessarily mean that Shimla's temperature is always the representative value, but that it represents the average value. The average here is a number that represents a typical or central value within a set of data. It is calculated as the sum of all values divided by the number.
What is a Geometric Sequence and how does it work?
A geometric sequence is a sequence of numbers in which consecutive terms have a common ratio. The sequence is considered geometric if it is multiplied or divided with the same non-zero number.
This progression can be represented as a, ar2, ar3, and so forth (where a refers to the first term and r the common ratio).
Example: 3, 9, 27, 81
The exponential form of the geometric sequence can be expressed as tn = 1. r(n-1) (where tn, t1 and d are the terms that make up the common ratio).
Although they are more difficult to understand than the arithmetic means, geometrical sequences still have many uses in daily work, such as in the calculation of growth rates, stock market prices, and interest rates.
Each term in a geometric sequence is equal to the preceding term times a constant multiplier, known as the common factor. Geometric sequences may have a fixed number or an infinite number of terms. The terms of a geometric sequence can quickly become very large, very small, or very close to zero in either one of these cases. The terms of a geometric sequence can change more quickly than arithmetic ones, but infinite arithmetic patterns increase or decrease continuously, while geometric sequences can approach zero depending on the common factor.
Geometric Sequence Properties
The geometric mean of geometric sequences has special properties. The square root of a product is the geometric mean of two numbers. The geometric mean of 5 and 20, for example, is 10, because 5 x 20 = 100. The square root of 100 is 10.
Each term in a geometric sequence is the geometric mean of each term that precedes it and the term that follows it. In the above sequences 3, 6, 12,..., 6 is the geometric mean of 3 and 12, while 12 is the geometry mean of 6 and 24, while 24 is the mean of 12 and 48.
The common factor is a key factor in other properties of geometric sequences. The common factor r must be greater than 1. Infinite geometric sequences will reach positive infinity if it is greater than 1. The sequences will be close to zero if r is between 0 & 1. If r is between 0 and 1, the sequences will approach zero. However, terms will alternate between negative and positive values. If r is lower than -1, terms will tend toward positive and/or negative infinity. They alternate between negative and positive values.
Geometric sequences, and their properties, are particularly useful for mathematical and scientific models of real-world processes. Specific sequences are useful for studying populations that grow at a certain rate over time, or investing that earns interest. It is possible to predict the future using both general and recursive methods, which are based on the common factor and the starting point.
How do you Distinguish and find the Difference Between Arithmetic Mean and Geometric Sequence?
These points are crucial in determining the difference between the arithmetic mean of the geometric sequence.
- Arithmetic means are a group of numbers where each phrase differs from its predecessor by a fixed amount. A geometric sequence is a collection of integers where each element is obtained by multiplying the preceding number by constant factors.
- A series can be considered arithmetic if there is a common term between the terms that follow. This is represented by 'd'. A sequence is considered geometric if there is a common ratio among succeeding terms (refer to 'r').
- Arithmetic means a way to find a new term by subtracting or adding a fixed value to the previous term. Contrary to geometric sequence, the new value is obtained by multiplying or subtracting a fixed value from previous terms.
- The variation in an arithmetic sequence's members is linear. The variation between the elements of an arithmetic sequence is, however, linear.
- Infinite arithmetic means diverge, but infinite geometric sequences converge and diverge depending on the situation.
Main Difference Between Arithmetic Mean and Geometric Sequence in Points
- The arithmetic average is the average of the terms collected by dividing the total number of terms by the number of given ones, while geographic sequence refers to the sequence of consecutive terms that have a common ratio.
- You can get the arithmetic average by adding the terms to the collection and dividing them by their number. The geometric sequence can be obtained by multiplying the constant non-zero term with the preceding number.
- The geometric sequence has exponential variation, while the arithmetic means is the central tendency in the data set.
- The arithmetic means is often used in experiments, observational studies, and other types of research, whereas the geometric sequence is used in stock markets to calculate savings, costs, and so on.
- The arithmetic means is used to reduce large data to give an approximate idea of the results. Geometric sequence, on the other hand, is the sequence of exact data. Therefore, the 'arithmetic average' can not always give exact results.
Conclusion
The arithmetic average is the average of a group of numbers, where the common difference among successive terms may or not be defined using a constant. However, the Geometric Sequence simply refers to the sequence of terms in which successive terms have a common ratio defined with 'r'.
The arithmetic means is calculated by multiplying the sum of all terms by the number of terms in the series. However, the geometric sequence is obtained by multiplying or dividing successive terms using the common ratio.
The arithmetic median is often the central limit of any data, while the geometric sequence is the exponential increase in the given collection.
If we look around, both the geometric sequences and the arithmetic means can be used in our daily lives. The arithmetic means is used in many fields, including anthropology and experimental studies to determine an average value, whereas the geometric sequences are used for calculating population growth, stock market prices, and other such things.
References
- https://www.tandfonline.com/doi/abs/10.1080/00029890.2001.11919815
- https://www.fq.math.ca/Scanned/22-4/schoen.pdf