Arithmetic Sequence vs Geometric Sequence
The study of patterns of numbers and their behaviour is an important study in the field of mathematics. Often these patterns can be seen in nature and helps us to explain their behaviour in a scientific point of view. Arithmetic sequences and Geometric sequences are two of the basic patterns that occur in numbers, and often found in natural phenomena.
The sequence is a set of ordered numbers. The number of elements in the sequence can either be finite or infinite.
More about Arithmetic Sequence (Arithmetric Progression)
An arithmetic sequence is defined as a sequence of numbers with a constant difference between each consecutive term. It is also known as arithmetic progression.
Arithmetic Sequnece ⇒ a1, a2, a3, a4, …, an ; where a2 = a1 + d, a3 = a2 + d, and so on.
If the initial term is a1 and the common difference is d, then the nth term of the sequence is given by;
an = a1 + (n-1)d
By taking the above result further, the nth term can be given also as;
an = am + (n-m)d, where am is a random term in the sequence such that n > m.
The set of even numbers and the set of odd numbers are the simplest examples of arithmetic sequences, where each sequence has a common difference (d) of 2.
The number of terms in a sequence can be either infinite or finite. In the infinite case (n → ∞), the sequence tends to infinity depending on the common difference (an → ±∞). If common difference is positive (d > 0), the sequence tends to positive infinity and, if common difference is negative (d < 0), it tends to the negative infinity. If the terms are finite, the sequence is also finite.
The sum of the terms in the arithmetic sequence is known as the arithmetic series: Sn= a1 + a2 + a3 + a4 + ⋯ + an = ∑i=1→n ai; and Sn = (n/2) (a1 + an) = (n/2) [2a1 + (n-1)d] gives the value of the series (Sn).
More about Geometric Sequence (Geometric Progression)
A geometric sequence is defined as a sequence in which the quotient of any two consecutive terms is a constant. This is also known as geometric progression.
Geometric sequence ⇒ a1, a2, a3, a4, …, an; where a2/a1 = r, a3/a2 = r, and so on, where r is a real number.
It is easier to represent the geometric sequence using the common ratio (r) and the initial term (a). Hence the geometric sequence ⇒ a1, a1r, a1r2, a1r3, …, a1rn-1.
The general form of the nth terms given by an = a1rn-1. (Losing the subscript of the initial term ⇒ an = arn-1)
The geometric sequence can also be finite or infinite. If the number of terms are finite, the sequence is said to be finite. And if the terms are infinite, the sequence can either be infinite or finite depending on the ratio r. The common ratio affects many of the properties in geometric sequences.
r > o
|0 < r < +1|
The sequence converges – exponential decay, i.e. an → 0, n → ∞
|r = 1|
Constant sequence, i.e. an = constant
|r > 1|
The Sequence diverges – exponential growth, i.e. an → ∞, n → ∞
r < 0
|-1 < r < 0|
The sequence is oscillating, but converges
|r = 1|
The sequence is alternating and constant, i.e. an = ±constant
|r < -1|
The sequence is alternating and diverges. i.e. an → ±∞, n → ∞
|r = 0|
The sequence is a string of zeros
N.B : In all the cases above, a1 > 0; if a1 < 0, the signs related to an will be inverted.
The time interval between the bounces of a ball follows a geometric sequence in the ideal model, and it is a convergent sequence.
The sum of the terms of the geometric sequence is known as a geometric series; Sn = ar+ ar2 + ar3 + ⋯ + arn = ∑i=1→n ari. The sum of the geometric series can be calculated using the following formula.
Sn = a(1-rn )/(1-r); where a is the initial term and r is the ratio.
If the ratio, r ≤ 1, the series converges . For an infinite series, the value of convergence is given by Sn = a/(1-r)
What is the difference between Arithmetic and Geometric Sequence/Progression?
• In an arithmetic sequence, any two consecutive terms have a common difference (d) while, in geometric sequence, any two consecutive terms have a constant quotient (r).
• In an arithmetic sequence, the variation of the terms is linear, i.e. a straight line can be drawn passing through all the points. In a geometric series, the variation is exponential; either growing or decaying based on the common ratio.
• All infinite arithmetic sequences are divergent, whereas infinite geometric series can either be divergent or convergent.
• The geometric series can show oscillation if the ratio r is negative while the arithmetic series does not display oscillation