Arithmetic sequence can be defined as an expression of terms arranged in a particular pattern such that difference between two consecutive terms is constant. That is an Arithmetic Progression is defined as a sequence or series of Numbers formed by addition of constant term to each term of expression. It follows a pattern like next term is obtained when a constant term is added to that term and this constant term is called as Common Difference. Every series consists of a first term and a common difference for finding different terms and parameters. But, if we want to find the first term of an arithmetic sequence we must know different parameters for that. For finding first term of an Arithmetic Series we must know value of common difference of series. Let’s take different formulas, formula for determining n th term of series that is an = a + (n - 1) d.
If in this above expression last term that is n th term is given and number of terms are given then we can easily determine first term of series by modifying this expression to a = an – (n - 1) d.
When we don't know value of common difference but sum and last term of series is given and number of terms are also given then first term of sequence can be determined by formula Sn = n / 2 [ a + l]. Modifying this formula for determining first term that is ‘a’ in equation we get “a = [(Sn * 2) / n] – l”. Here, ‘a’ is first or initial term, Sn is sum of 'n' terms, 'n' is number of terms and 'l' is last term of arithmetic series. In this expression we do not use common difference.
When last term is not given and sum is given then we will find last term using sum of terms formula and then use last term in the last term formula for determining first term.rogression is defined as series formed by multiplication of a constant term to obtain next term of series. Here this constant term is as common ratio. Ratio of two consecutive terms in a Geometric Sequence is constant which is defined as common Ratio and represented by 'r'. Sum of Geometric Progression is termed as Geometric Series. This series includes terms in a geometric progression. As we know that progression refers to increment or process of any identity in any particular format or in any specified pattern. Sum of Geometric Progression involves sum of finite terms of series as well as sum of infinite terms. The n th term of a geometric series is determined by the expression an = ar n – 1 where 'a' is first or initial term and 'r' is common ratio between two terms of series. The series which follows a recursive relation is represented as an = r an – 1. It is defined for Integer whose value is greater than or equal to 1. Sum of a geometric progression when series start from initial value is given as:
Sn = a (1 – rn+1) / (1 - r).
And when initial value does not starts from zero that is it has some value for initial term which is represented as:
Sn = a (rm – rn+1) / (1 - r) where 'm' is degree of initial term when it is not zero.
Sum of infinite series of a geometric progression is represented as
Sn = a / (1 - r).
For example: Consider a series 2, 4, 8, 16. . . . . . . . . . up to infinity. Here, initial term or first term ‘a’ is equals to 2 and common ratio that is ‘r’ is equals to 2. Then sum of this series is determined by
S∞ = a / (1 - r).