Question

What is the difference between Arithmetic and Geometric progression?

Answer 1

First, we need to know about the concept of Arithmetic and Geometric progression.
An arithmetic progression can be given by $a,(a + d),(a + 2d),(a + 3d),...$ where $a$ is the first term and $d$ is a common difference.
A geometric progression can be given by $a,ar,a{r^2},....$ where $a$ is the first term and $r$ is a common ratio.

Complete step by step answer:
The arithmetic progression can be expressed as ${a_n} = a + (n - 1)d$
Where $d$ is the common difference, $a$ is the first term, since we know that difference between consecutive terms is constant in any A.P
For GP the formula to be calculated ${S_n} = \dfrac{{a({r^n} - 1)}}{{r - 1}},r \ne 1,r > 1$ and ${S_n} = \dfrac{{a(1 - {r^n})}}{{1 - r}},r \ne 1,r < 1$
Difference between the AP and GP:
Arithmetic Progression:
The series is defined as the new terms in the difference between two consecutive terms so that they have a constant value.
In AP the series is identified with the help of a common difference between the two consecutive terms.
The AP terms are varying as in the form of linear (degree one)
Geometric Progression:
New series are obtained by multiplying the two consecutive terms so that they have constant factors.
In GP the series is identified with the help of a common ratio between consecutive terms.
Series vary in the exponential form because it increases by multiplying the terms.

Note: Relation between AM, GM, and HM can be expressed as $G.{M^2} = A.M \times H.M$
Harmonic progress is the reciprocal of the given arithmetic progression which is the form of $HP = \dfrac{1}{{[a + (n - 1)d]}}$ where $a$ is the first term and $d$ is a common difference and n is the number of AP.
For AP $a,b,c$ are said to be in arithmetic progression if the common difference between any two-consecutive number of the series is the same that is $b - a = c - b \Rightarrow 2b = a + c$