Arithmetic Progression vs Geometric Progression

Arithmetic progression and Geometric Progression are two types of mathematical progressions which are generally used to determine a series of Numbers. arithmetic progression vs geometric progression is best defined by the way they generate the series. A Geometric Sequence is the one in which each number in the series is multiplied by a constant factor to generate next number. Whereas an Arithmetic Progression is a sequence in which each number is generated by adding a common difference to preceding number. General representation of a geometric progression with first term as 'A' and common factor as 'R' is given as:
A, A * R, A * R², A * R³ and so on,
Whereas representation for an arithmetic progression with 'A' as first term and’d’ as common difference is given as:
A, A + d, A + 2d, A+ 3d and so on
Where,
A (s + 1) = A (s) + d, A (s) and A (s + 1) are the sth and s+1th terms.
Or A (s + 1) - A (s) = d
Example to show Arithmetic vs geometric sequences difference can be: Geometric Sequence given as5, 25, 125, 625…. with the first term as 5 and common factor given as 25 / 5 or 625 / 125 = 5 and Arithmetic Progression given as 5, 9, 13, 17, 21... with 5 as first term and 4 as common difference.
So, in an arithmetic progression we just need to know first term and common difference or the first term and the succeeding term or any random term with its Position in AP to generate the complete sequence. Whereas in a geometric progression we need to know the first term and the common factor or the first term and the succeeding term or any random term with its position in GP to generate the complete sequence. The formula to calculate the first term when any term of the sequence is given with its position is also different for both the progressions.
For a geometric progression it is given as:
a = s / [F (t – 1)],
Where, s represents the random term of the sequence with its position in the sequence as t and the common factor as F. So if the 5th term in the sequence is 625 and the common Ratio is 5 we can write, s = 625, t = 5 and F = 5. Then n can be calculated as
n = 625 / [5(5 - 1)] = 1.
So, complete sequence can be defined as: 1, 5, 25, 125, 625…….
Whereas for an arithmetic progression the formula is given as:
A (N) = A (1) + N *d,
Where, 'n' represents position of term A (N) in the series, 'D' is common difference. If 'N' is total number of terms in series A (1) and A (N) represents the first and last terms in series.