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What is Arithmetic Sequence & Arc Length. Their Importance?

A list of numbers having a specific pattern such that the difference between consecutive terms is always constant is known as an arithmetic sequence. In other words, we can say that in a sequence we always add the same value or number (constant) repeatedly for the infinite times.

Arithmetic Sequence

 In arithmetic sequence every successive value is formed by subtracting or by addition of a definite number to the previous numbers. For instance, in arithmetic sequence if we choose a number from the list and minus it from the preceding one then the answer will be constant or same. 

If we need to know or find out any value from the sequence, then the arithmetic sequence formula is used in the arithmetic sequence to find that term or the “nth” term. The arithmetic sequence can be defined by the two formulas:

• The Recursive Formula

• The Explicit Formula

The constant in the arithmetic sequence is known as common difference. We can define common difference as the constant difference between two consecutive terms in an arithmetic sequence. The common difference is indicated by “d” in the formula.

Example

For example, in an arithmetic sequence if the “d” or common difference is 6, then it implies that in every consecutive or successive sequence 6 is added. And if the common difference is a negative value, let’s suppose -5, this implies in every sequence we have to subtract 5.

Arithmetic Sequence of number may calculate by using online arithmetic sequence formula calculator.

This simply explain that common difference is used to move from one term to another. To generate the values in the sequence we just have to add the common difference in our current value and keep on repeating the procedure.

The point to be note here, however, is the sequence is meant to be increasing if the common difference between two consecutive terms is positive. While the sequence is taken decreasing in case the common difference between consecutive value is positive.

Arc Length

The measure of distance along a curved path contributing the arc is known as the arc length. It is always longer than the distance between its end points or the straight line chord. Arc-length can also be defined as portion of a circumference or the distance between two points at the edge of the circle. 

The arc length is one of the applications of definite integrals and is used to measure the distance along a curve path. There are several implications of arc length in everyday life. For instance, if you are driving along a curve road the length of road is calculated from the concepts of arc length.

Real life Application

Or if we need to measure the distance travelled by the rocket if it is launched in a parabolic pathway we again use arc length formula to calculate its distance. The arc length of a circle is always proportional to the measure of central angle.

This means that if the central angle increases in measurement the arc length increases as well. Also, if the arc will decrease as a result of decrease in central angle measurement. In calculus this concept is quite helpful in finding the area of things other than circle as well.

Example

On the basis of central angle unit of the arc, arc length is calculated through various formulas. The central angle measurement may be provided in radians or degrees and thus we can accordingly calculate the arc length.

The arc length formula is θ times the radius of a circle. In radians the arc length formula is expressed as

arc length = θ × r, 

L = Arc Length

θ = Central angle of arc

r = Radius of the circle

Length of arc can calculate by different formula’s and calculations. However some notable tools i.e. arc length calculator are available that help us to perform such type of calculation easily.

when the radius and central angle are given, the central angle is not present, even the radius is not giving

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