Sequences and Series Formulas
Sequence and Series Formulas relate to the simple number series in math. 1, 2, 3, .... is the simplest example of arithmetic number series. This series has a regular pattern and it relates one term to the subsequent terms in the series. An example of geometric series is 2, 4, 8, 16, 32, ..... A harmonic series is of the form of 1, 1/2, 1/3, 1/4,....... Here the sequence and series formulas define the number relationship across each of the terms of these series.
What is the List of Sequences and Series Formulas?
The below list includes formulas for the arithmetic series, geometric series, and harmonic series. Here the sequence and series formulas include formulas to find the n^{th} term and to find the sum of the n terms of the series. In an arithmetic series, there is a common difference between two subsequent terms and in a geometric series, there is a common ratio between consecutive terms.
Formula 1:
The n^{th} term of an Arithmetic Series is as follows. Here a is the first term and d is the common difference between two consecutive terms. The number of d's in each term is one lesser than the term. There are (n  1) d's in the n^{th} term of the series.
Arithmetic Series: a, a+d, a+2d, a+3d, a+4d, .......a +(n1)d
a_{1} = a, a_{2} = a + d, a_{3} = a + 2d, a_{n} = a + (n  1)d
Formula 2:
The sum of n terms of an arithmetic series is as follows. The sum of the n terms is equal to the product of n/2 and the sum of the first term(a) and the last term (a + (n  1)d) of the arithmetic series.
S_{n} = n/2 ×(First Term + Last Term)
\[S_n = \frac{n}{2}.(2a + (n  1).d)\]
Formula 3:
The n^{th} term of a geometric series is as follows. Here a is the first term and r is the common ratio. The power of r is one lesser than the term of the geometric series.
Geometric Series: a, ar, ar^{2}, ar^{3}, ar^{4}, ar^{5}, ........ar^{n  1}
a_{1} = a, a_{2} = ar, a_{3} = ar^{2},
āān^{th} term of the Geometric Series: a_{n} = ar^{n  1}
Formula 4:
The sum of terms of a geometric series is obtained for the sum of n terms and for the sum of infinite terms of the series.
Sum of n terms of a Geometric Series: \(S_n = a\frac{(r^n  1)}{(r  1)}\)
Sum of infinite terms of a Geometric Series: \(S_n = \frac{a}{1  r}\)
Formula 5:
The n^{th} term of a harmonic series is as follows. The harmonic series is reciprocal of the arithmetic series. Here also d is a common difference and there is (n  1) number of d's in the nth term of the series.
Harmonic Series: 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d),..........
nth term of the Harmonic Series: \[a_n = \frac{1}{a +(n  1)d}\]
Let us check a few examples to explore the different sequences and series formulas.
Solved Examples on Sequences and Series Formulas

Example 1: Find the value of the 25^{th} term of the arithmetic series 5, 9, 13, 17.....
Solution:
The given series is 5, 9, 13, 17.....
First Terms a = 5
Common Difference d = 9  5 = 4
The 25^{th} term = T_{25} = a + 24d = 5 + 24*4 = 5 + 96 = 101
Answer: Hence the 25^{th} term of the series is 101. 
Example 2: The fifth term of the arithmetic series is 23, and the seventh term of the series is 31. Find the first three terms of the series.
Solution:
Given fifth term T_{5 } = 23, and T_{7} = 31.
a + 7d = 31
a + 5d = 23
By taking the difference of the above two terms we have:
2d = 8,
d = 4 and a = 7
Hence we have a = 7, a + d = 11, A + 2d = 15
Answer: Therefore the first three terms of the series is 7, 11, and 15.