In the world of mathematics there are many terms that are important in this and other areas, as well as being widely used in everyday life situations. One of these terms is sequences, which are very useful tools when describing number patterns and model phenomena that occur in real life.
On the other hand, there are quadratic sequences, which are those generated by an algebraic expression of the second degree, also known as quadratic equations. Throughout this article we will talk about the quadratic sequence specifically, its characteristics, how it is formulated, some examples of it in mathematics, and more.
What does quadratic sequence mean?
When we talk about a quadratic sequence, it is understood that it is a numerical sequence where each term is obtained from the previous terms by means of a quadratic formula. This results in a quadratic equation, whose general formula for its quadratic sequence is written as follows: an = ax² + bx + c
From this formula we can extract that “a”, “b” and “c” are constants, while “n” represents the term of the sequence that is being calculated. Likewise, each existing term in the sequence is calculated by substituting “n” in each equation and solving for “an”.
For his part, he limit of a quadratic sequence approaches infinity if the square coefficient is positive. Furthermore, this type of sequence can be interpreted as the evaluation of a quadratic function on the natural numbers.
In general, quadratic sequences tend to be widely used in various areas of mathematics, such as in engineering and physics. In addition, it can have many practical applications in the modeling and prediction of natural or economic phenomena.
What are the characteristics of a quadratic sequence?
quadratic sequences They are known for their different characteristics, which distinguish them from the rest of successions. That is why, below, we will describe some of the most important characteristics:
- The inference rules They allow us to find the general formula of a quadratic sequence starting from its terms.
- The general formula for a quadratic sequence is written as follows: an = ax² + bx + c. Where a, b and c are mathematical constants and n represents the term of the sequence being calculated.
- To determine the set of values of ‘n’ of a quadratic sequence, a quadratic inequality.
- Each one of the terms of the sequence is calculated by substituting “n” in the quadratic equation and solving for “an”. That is why, the general formula of the sequence allows us to calculate any term starting from the constants a, b and c.
- In technical language, it is understood that the graphic representation and the behavior of a quadratic sequence is a parabola. This means that the terms can either increase or decrease rapidly, which depends on the values of the constants a, b, and c.
- The vertex of the parabola is the one that defines the minimum or maximum value of the sequencesince this depends on the sign of the constant “a”.
- The value of “a” is also what determines the concavity of the parabola. This means that if “a” is positive, the parabola opens upwards and the sequence has a minimum value, but if it is negative, it opens downwards and the sequence will have a maximum value.
In short, quadratic sequences are known as a type of number sequence given by a quadratic formula and that it has a parabolic behavior. For their part, the named characteristics are what make quadratic sequences very useful in different areas of physics, engineering and mathematics in general.
What is the formula for quadratic sequences?
The general formula of a quadratic sequence It is written as follows: an = ax² + bx + c. Where “a”, “b” and “c” are constants, while “n” represents the term of the sequence to be calculated.
By using this formula we can calculate any term of the sequence from the constants, named above, ‘a’, ‘b’ and ‘c’. On the other hand, to find the specific values of these constants for a given quadratic sequence, two terms of the sequence and their corresponding positions are necessary.
Thus, to solve for the constants ‘a’, ‘b’ and ‘c’ we can use a system of equations. For example, if we have the terms a1 and a2 of a quadratic sequence, a system of equations can be written as follows: a1 = a + b + c a2 = 4a + 2b + c.
Then, solving the system of equations, you can find the specific values of ‘a’, ‘b’ and ‘c’, for this quadratic sequence. Once the values of the constants are known, we can calculate any term found in the sequence using the general formula.
How is a quadratic sequence made?
If we wish to create a quadratic sequence, we need to find a formula that relates each of the terms of the succession with the previous ones. When we talk about quadratic sequences, the formula would have the following form: an = ax² + bx + c.
Where the constants ‘a’, ‘b’ and ‘c’ are determined from the first two terms of said succession. As we mentioned in the previous point, we can use a system of equations to find the values of these constants.
Once we find the values of the constants, the general formula can be used to calculate any term of the sequence. Also, also You can graph the terms of the sequence on a Cartesian plane.where the horizontal axis represents the term number of the sequence in question, and the vertical axis represents the value of the term.
By plotting on a graph the points that correspond to the terms of the sequence, it can be observed how the parable of the succession behaves and the position of its vertex. This is the point in the sequence where the value can be maximum or minimum.
In general, it is understood that, to create a quadratic sequence, it is required to find a formula that relates the terms of said sequence with the previous ones, using a quadratic equation. Thus, the values of each term can be calculated from said formula.
Examples of quadratic sequences in mathematics
There are various examples of the use of quadratic sequences, both in mathematics and in daily life and in other sciences. That is why we will describe some examples below:
- For him calculation of the acceleration of a ton in free fall: the quadratic sequence that describes the position of a ton in free fall is given by the expression h(n) = -4.9n^2 + 50, where h is the height in meters and n is the time in seconds.
- The sequence of the following perfect squares: 1, 4, 9, 16, 25, 36… has the quadratic formula an = n^2.
- In mathematics, one finds the sequence of triangular numbers such as the following: 1, 3, 6, 10, 15, 21… These have the quadratic formula an = (1/2)n^2 + (1/2)n.