An arithmetic series consists of consecutive numbers with the difference being a constant. If this is true then the relationship given below is valid:
Sn = a1 + a2 + a3 + a4 +⋯+ an = ∑ni=1 ai ; where a2 = a1 + d, a3 = a2 + d, and so on.
In the equation above, a1 is the first term, d if the constant difference
The nth term is given by the equation an = a1+ (n-1)d
It should be noted that the behaviour of the series largely depends on the common difference. An increase or decrease in the common difference can make the progression either negative infinite or positive infinite. The formula given below is commonly used to calculate the sum of the series. This formula was developed by famous mathematician and astronomer Aryabhata.
Sn = n/2 (a1+ an ) = n/2 [2a1 + (n-1)d]
Where the Sum Sn can be either finite or infinite depending on the numbers involved in the set and the common difference.
- Image courtesy: verjinschi.disted.camosun.bc.ca
For a geometric series, the quotient of the consecutive number must be a constant value. According to research conducted by scientists and mathematicians, geometric series can play a key role in solving a variety of engineering problems, especially due to the properties it possesses.
Sn = ar + ar2 + ar3 +⋯+ arn = ∑ni=1 ari
The sum of this type of series can be obtained by using the formula given below:
Sn = a(1-rn) / (1-r)
Where r is the ratio
- Image courtesy: astronomy.mnstate.edu