Question

What is the difference between Arithmetic and Geometric progression?

Answer 1

Hint: 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 $