Sequences and Series Formulas
Sequence and series formulas are related to different types of sequences and series in math. A sequence is the set of ordered elements that follow a pattern and a series is the sum of the elements of a sequence. The sequence and series formulas generally include the formulas for the n^{th} term and the sum.
What are Sequences and Series Formulas?
The below list includes sequences and series formulas for the arithmetic, geometric, and harmonic sequences. Here the sequence and series formulas include formulas
- to find the n^{th} term of the sequence and
- to find the sum of the n terms of the series.
In an arithmetic sequence, there is a common difference between two subsequent terms. In a geometric sequence, there is a common ratio between consecutive terms. In a harmonic sequence, the reciprocals of its terms are in an arithmetic sequence. The figure below shows all sequences and series formulas.
Let us see each of these formulas in detail and understand what each variable represents.
Arithmetic Sequence and Series Formulas
Consider the arithmetic sequence a, a+d, a+2d, a+3d, a+4d, ...., where 'a' is its first term and 'd' is its common difference. Then:
- n^{th} term of arithmetic sequence, a_{n} = a + (n - 1) d
- Sum of the arithmetic series, S_{n} = n/2 (2a + (n - 1) d) (or) S_{n} = n/2 (a + a_{n})
Geometric Sequence and Series Formulas
Consider the geometric sequence a, ar, ar^{2}, ar^{3}, ..., where 'a' is the first term and 'r' is the common ratio. Then:
- n^{th} term of geometric sequence, a_{n} = a r^{n - 1}
- Sum of the finite geometric series (sum of first 'n' terms), S_{n} = a (1 - r^{n}) / (1- r)
- Sum of infinite geometric series, S_{n} = a / (1 - r) when |r| < 1. The sum is not defined when |r| ≥ 1.
Harmonic Sequence and Series Formulas
Consider the harmonic sequence 1/a, 1/(a+d), 1/(a+2d), 1/(a+3d), 1/(a+4d), ...., where '1/a' is its first term and 'd' is the common difference of the arithmetic sequence a, a + d, a + 2d, .... Then:
- n^{th} term of harmonic sequence, a_{n} = 1 / (a + (n - 1) d)
- Sum of the harmonic series, S_{n} = 1/d ln [ (2a + (2n - 1) d) / (2a - d) ]
Examples on Sequences and Series Formulas
Example 1: Find the value of the 25^{th} term of the arithmetic sequence 5, 9, 13, 17.....
Solution:
The given sequence is 5, 9, 13, 17.....
The first term, a = 5
The common difference, d = 9 - 5 = 4
Using the sequence and series formulas,
a_{n} = a + (n - 1) d
For the 25^{th} term, substitute n = 25:
a_{25} = a + 24d = 5 + 24*4 = 5 + 96 = 101
Answer: Hence the 25^{th} term of the series is 101.
Example 2: Find the sum of the first 100 terms of the arithmetic series 1 + 4 + 7 + ....
Solution:
In the given series, the first term is a = 1 and the common difference is d = 3.
Using the sequences and series formulas,
S_{n} = n/2 (2a + (n - 1) d)
For the sum of 100 terms, substitute n = 100:
S_{100} = 100/2 (2(1) + (100 - 1) 3) = 14,950
Answer: The sum of the first 100 terms is 14,950.
Example 3: Is it possible to find the sum of the infinite geometric series 5, -5/2, 5/4, -5/8, ...? If so, find the sum.
Solution:
In the given geometric series, the common ratio, r = -1/2. So |r| = |-1/2| = 1/2 < 1. So it is possible to find its sum using one of the sequence and series formulas:
S = a / (1 - r)
= 5 / (1 - (-1/2))
= 5 / (3/2)
= 10/3
Answer: Sum of all terms of the given series = 10/3.
FAQs on Sequences and Series Formulas
List some Important Sequences and Series Formulas.
The sequences and series formulas for different types are tabulated below:
Arithmetic | |
Sequence formula of the n^{th} term | a_{n} = a + (n - 1) d |
Series formula for the sum of n terms | S_{n} = n/2 (2a + (n - 1) d) |
Geometric | |
Sequence formula of the n^{th} term | a_{n} = a r^{n - 1} |
Series formula for the sum of n terms | S_{n} = a (1 - r^{n}) / (1- r) |
Series formula for the sum of infinite terms | S = a / (1 - r) when |r| < 1 |
Harmonic | |
Sequence formula of the n^{th} term | a_{n} = 1 / (a + (n - 1) d) |
Series formula for the sum of n terms | S_{n} = 1/d ln [ (2a + (2n - 1) d] / (2a - d) ] |
What is the Difference Between Sequence and Series Formulas?
The sequence formulas would tell how to find the n^{th} term (or general term) of a sequence whereas the series formulas would tell us how to find the sum (series) of a sequence.
When to Use Sequences and Series Formulas?
To find the n^{th} term of a specific sequence, use the sequence formulas. To find the sum of terms of a sequence, use the series formulas.
What is the Sum of a Harmonic Series Using Sequences and Series Formulas?
For a harmonic sequence 1/a, 1/(a+d), 1/(a+2d), 1/(a+3d), 1/(a+4d), ...., the sum of its first 'n' terms can be found using the formula S_{n} = 1/d ln [ (2a + (2n - 1) d] / (2a - d) ].
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