Arithmetic (algebraic) sequence - definition, examples of finding with solution

Often, when solving problems related to observations and assigning a value to a particular event over a certain period of time, a series of numbers is obtained, which is called an arithmetic sequence.

Table of contents:

• Definition and examples of arithmetic sequence

• Types of arithmetic (algebraic) sequence

• Formulas of arithmetic sequence

• Examples of problems with solutions.

One of the main distinctive features - such a mathematical model has a regularity, based on which any unknown term can be calculated, which simplifies forecasting when calculating physical situations.

Examples of everyday use may include monitoring air temperature, predicting expenses by entering results into a table, and more.

Definition and examples of arithmetic sequence

This is a sequence of numbers, where each subsequent number in the series (starting from the second) increases or decreases by a certain amount, which is a constant.

a₁, a₁+d, a₁+2d,..., a₁+(n-1)d,...

In addition, a number of related terms and definitions are used to describe. A member (an) is a single number from a sequence.

Difference (d) is a fixed number by which the next progression number increases or decreases.

aₙ₊₁=aₙ+d

In addition, there are types of such series:

• increasing - the numbers of the series increase in value

• decreasing - each subsequent number of the series decreases.

As an example, imagine the sequence of numbers "3, 9, 15, 21, 27". This case - this series of numbers falls under the characteristics of an arithmetic progression. This conclusion is made when the difference between the terms of the series is fixed and equals 6.

Types of arithmetic (algebraic) progression

Varieties are built on the basis of the difference characteristic (d), namely, on the basis of the difference of the latter from zero.

Thus, you can meet certain variations:

• difference d0 - this assumes that each term in the series will be greater than the previous one, and the progression will be called increasing

• with d=0, the series will also have the properties of a progression, which is called stationary, and all members will be the same (they will not change).

If the progression does not change with each step by the same difference, then this progression is not constant and is not arithmetic.

Note: the arithmetic differs from the geometric one in that in the latter, each subsequent one is increased by the same factor.

Formulas of arithmetic progression

One of the most important properties is the ability to calculate any number of a specific place in a series.

• Definition:aₙ₊₁ = aₙ + d
• Difference: d = aₙ₊₁ - aₙ
• n-th term formula aₙ = a₁ + (n-1)d
• sum of n-th terms formula

  Sₙ = n(a₁ + aₙ)/2

To solve this, you need a formula showing how the term of an arithmetic progression is found. In general, it will look like the value of the previous number in the series (an-1), to which the difference (d) is added:

aₙ = aₙ₋₁ + d

A problem may also arise when it is necessary to sum all the numbers of an arithmetic progression series (the sum of terms). If there are a small number of them, then you can calculate it manually, but if the number of numbers exceeds a hundred, then it will be easier to use a special formula for processing.

Sₙ = n(a₁ + aₙ)/2

So, we need the value of the first number in the series (a1) and the last one (an), as well as information about the total number of numbers in the series. The recursive formula showing how to find the sum would then look like this:

Note: the value n means exactly the number of members of the series for which the sum is found.

The product of the terms of an arithmetic progression can be found using a similar formula:

Pₙ = (b₁bₙ)ⁿ/²

where, Pn is the product, b1 and bn are the first and last numbers, respectively, and n is the number of terms.

Separately, one should touch on such a thing as a characteristic property of a progression. It boils down to fulfilling a certain condition for each element:

aₙ = (aₙ₋₁ + aₙ₊₁)/2, n>= 2

Examples of problems with solutions.

Consider how to solve problems on a given topic.

Example 1

What is the 574th number in an arithmetic sequence where the first three numbers are 8, 15, and 22?

One way to solve Example 1 is to use the formula for finding a specific term in an arithmetic sequence. To use this formula, we need to know the value of the first term (a1) and the common difference (d) between the terms. To find the common difference, we can subtract the first term from the second term (15 - 8) to get d = 7. With this information, we can now use the formula to find the specific term we're interested in.

aₙ = a₁ + (n-1)d

Substituting the obtained values, we get an expression like a_574 = 8 + (574-1) * 7.

After the calculation, we get the answer: a_574 = 4019.

Example 2

To prepare for the biology exam, a student needs to learn 730 questions (including riddles). He is known to be quite anxious, and as the exam date approaches, he learns 27 more questions a day than on the previous day. A student friend found out that the student only learned 17 questions on the first day.

It is required to find out how much time the student spent on preparation.

Solution: Obviously, the case of preparing a student for an exam is solved through formulas by an arithmetic progression (since there is a fixed difference d = 17). Substitution of known data:

aₙ = a₁ + (n-1)d

After substitution, we get the expression: 730 = 17 + (n - 1) * 27.

After calculations, we determine the answer - 27 days.

Arithmetic progression is the simplest of all numerical relationships. The use of the described formulas will greatly speed up calculations in tasks where it is required. In addition, to simplify, you can use the online calculator.