Concepts of Distance Formula & Arithmetic sequence in Linear Algebra

Distance Formula

An equation implied for determining a line segment’s length, given its ends, is known as distance formula. As the inputs for the formula of the distance are two points. The distance between two points can also be determined using the distance formula.

In order to calculate the distance between two points, that can be expressed randomly as points (x1,y1), and (x2,y2), the Distance Formula is a helpful tool. For length measurement of line segments and points in a two-dimensional space, the distance formulation is employed. 

Distance Formula as Pythagoras Theorem

The distance formula may appear complex and tricky to grasp. The best method to keep track of the plus and minus signs and squares and square roots, though, is to recall the derivation of the distance formula i.e. the Pythagoras Theorem.

The Distance Formula itself is based on the Pythagorean theorem, i.e. a2 + b2 = c2 where c is the longest side of a triangle also known as the hypotenuse and the other two side i.e. a and b are shorter sided known as the legs of a right triangle.

The basic concept of the Distance Formula is that the length of the right triangle’s hypotenuse, as expressed as letter c, is calculated. Therefore, the distance d between two given points (x1y1) and (x2y2), is calculated using the distance formula as given below

d=x2-x12+y2-y12  

How to Calculate Distance Formula

A line segment or two points can be calculated using the distance formula but how to calculate the distance? Using the equation above (distance formula) the step by step method of calculating distance between two points is mentioned below:

  • Coordinates Identification

First, we must identify the coordinates of the line segment’s ends or the two points in the given statement or figure.

  • X and Y Coordinates Representation

A point’s coordinates are (x, y), where x and y are integers indicating the horizontal and vertical distances from the origin respectively. A movement to the downside or left side is represented by a negative number. Contrarily, the movement to the right and side is represented by the positive numbers.

  • Check Grid Lines

Grid lines represent a set interval in coordinate planes. Both horizontal and vertical grid lines might differ. Before determining a point’s coordinates, check the grid line interval length.

  • Count Grid Lines

For the x-coordinate of a point, we count the vertical grid lines between it and the origin, then multiply that number by the grid line interval length. In the same way, the number of horizontal grid lines between origin and multiplied by the interval length, forms the y coordinate.

  • Putting the Values

Now choose one of the points, one as (x1, y1) and the other as (x2, y2). We may use the distance formula to find the distance between these two locations. Then putting their values in the formula.

But Beside to following all these steps, just put your question in distance formula calculator with variables. This will provide you the complete possible solution of that problem with steps.

Concept of Arithmetic Sequence

A sequence in which the consecutive terms are calculated by adding a constant value (positive or negative) to the preceding term. 

The constant value that is either added or subtracted from the previous value is termed as the common difference (d). The arithmetic sequence in mathematical terms is expressed using the formula

Tn = a+(n−1)d

In the expression above the nth term is represented as Tn, whereas in a sequence the n is the position of a term. Moreover, to represent the first term expression a is used while the is common difference as a constant is expressed as d.

Definition of Arithmetic Sequence

An ordered collection of numbers (called terms) is an arithmetic (or linear) sequence. Here each new term is calculated by adding a constant value to the preceding term. 

The recursive formulae, i.e. a1 = c, an+1 = an + d, where d again makes the common difference between successive terms and c is a constant, can be used in order to define the arithmetic sequence. For an infinite arithmetic series, its sum can either be ∞, if d > 0, or – ∞, if d < 0.

Related: You may also learn more educational blogs that are fully accurate and helpful for students only on Screenshot9.

Calculating the Sum of Arithmetic Sequence 

The sum of a finite arithmetic series can be found using two different methods. 

First Method

In order to utilize the first method of calculating arithmetic sequence, you should know the first term’s value a1 and the last term’s value an.  The first n terms of the arithmetic sequence are then summed as using the formula, Sn = n

Second Method

For the second method, you must know the first term a1 value and the common difference d. Then, according to Sn = na1 + n/2 (dn – d) the sum of the first n terms of arithmetic sequence can be calculated.

Common Difference

Since it is common to all subsequent terms pairs, it is referred to as the common differential. It’s represented with small letter “d”. Moreover, the sequence is not arithmetic if the difference between consecutive terms is not consistent. 

Two successive values of the sequence have to be subtracted in order to found their common difference. For the common difference of an arithmetic sequence the formula used is: 

d = an+1 – an

Also you may try free arithmetic sequence calculator with solution which will find the sequence of arithmetic series without your brain efforts.

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