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Explicit Formulas
Explicit formulas are helpful to represent all the terms of a sequence with a single formula. The explicit formula for an arithmetic sequence is a_{n} = a + (n  1)d, and any term of the sequence can be computed, without knowing the other terms of the sequence.
In general, the explicit formula is the n^{th} term of arithmetic, geometric, or harmonic sequence. Let us check the explicit formulas and how to find the explicit formulas of each of the sequence, with the help of examples, FAQs.
What are Explicit Formulas?
Explicit formulas are always used to represent any term of the sequence, without writing the other terms of the sequence. The meaning of "explicit" is direct, something that can be directly found without knowing the other terms of the sequence. The terms of a sequence can be uniquely represented using a single formula, which is the explicit formula. The explicit formula helps to easily find any term of the sequence, without knowing its previous term. Here are the explicit formulas of different sequences:
 Arithmetic Sequence: a_{n} = a + (n  1) d, where 'a' is the first term and 'd' is the common difference.
 Geometric Sequence: a_{n} = a r^{n  1}, where 'a' is the first term and 'r' is the common ratio.
 Harmonic Sequence: a_{n} = 1 / [a + (n  1) d], where 'a' is the first term and 'd' is the common difference of the arithmetic sequence formed by taking the reciprocals of the harmonic sequence.
The above explicit formulas are helpful to find any term of the arithmetic sequence, geometric sequence, or harmonic sequence, by simply substituting the n values in the respective explicit formulas.
Let us consider a simple sequence of even numbers(2, 4, 6, 8, .....) which can be uniquely represented using an explicit formula (a_{n} = 2n). Generally, the n^{th} term of the sequence represents the explicit formula. The explicit formula is also helpful to represent the entire sequence with a single formula.
How To Find Explicit Formulas?
The explicit formula can be computed from the n^{th } term of the given sequence. The n^{th }term of the arithmetic sequence is a_{n} = a + (n  1)d and plugging in the value of a and d gives the required explicit formula of the arithmetic sequence. Here 'a' is the first term of the sequence, and 'd' is the common difference of the arithmetic sequence. These are the three simple steps to find the explicit formula.
 First, find the first term and the common difference of the arithmetic sequence.
 Substitute the values in the n^{th} term of the explicit formula.
 Simplify the n^{th} term mathematically to get the explicit formula.
The explicit formulas can also be derived for the terms of the geometric progression and harmonic progression respectively.
Explicit Formula For Arithmetic Sequence
The arithmetic sequence explicit formula is used to find any term of the arithmetic sequence without calculating the previous term or any other term of the arithmetic sequence. The explicit formula is the nth term of an AP, and can be directly used to find any term of the arithmetic progression.
Explicit formula for finding the n^{th} term of arithmetic sequence: a_{n} = a + (n  1)d
Let us take an arithmetic sequence 4, 7, 10, 13, 16, ....... The first term of this sequence is a = 4, and the common difference is d = 7  4 = 3. Now we can easily find the arithmetic sequence explicit formula from the n^{th} term of the sequence. The n^{th} term of the arithmetic sequence is a_{n} = a + (n  1)d. The explicit formula is a_{n} = 4 + (n  1)3 = 3n  3 + 4, or we have a_{n} = 3n + 1.
Explicit Formula For Geometric Sequence
The geometric sequence explicit formula can be used to find any term of the geometric sequence. The explicit formula for the geometric sequence a, ar, ar^{2}, ar^{3}, ........ar^{n  1 }is its n^{th} term. The explicit formula for the geometric sequence is a_{n} = ar^{n  1} and it is also the nth term of a GP. Here 'a' is the first term of the geometric sequence, and 'r' is the common ratio of the geometric sequence. The common ratio formula is r = ar/a = ar^{2}/ar, and it is obtained by dividing a particular term with its previous term.
Explicit formula for finding the n^{th} term of geometric sequence: a_{n} = ar^{n  1}
Let us understand this by finding the explicit formula of a geometric sequence, 2, 6, 18, 54,... The first term of this sequence is a = 2, and the common ratio of this sequence is r = 6/2 = 3. The explicit formula for the geometric sequence is a_{n} = ar^{n  1} = 2.3^{n  1}.
Explicit Formula For Harmonic Sequence
The harmonic sequence explicit formula is useful to easily find any term of the harmonic sequence without finding the other terms of the sequence. The terms of the harmonic sequence are 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), .....1/(a + (n  1)d). Here 1/(a + (n  1)d) is the general term of the harmonic sequence and is the required explicit formula.
Explicit formula for finding the n^{th} term of harmonic sequence: a_{n} = 1/(a + (n  1)d)
This explicit formula of the harmonic sequence helps to easily find any term of the sequence, without knowing the previous terms. For the harmonic sequence 1/2, 1/5, 1/8, 1/11, .... the value of a = 2, and d = 5  2 = 3, and the nth term of the sequence is 1/(2 + (n  1)3) = 1/(3n  1). Thus the harmonic sequence explicit formula for this sequence is a_{n} = 1/(3n  1).
Important Notes in Explicit Formulas:
 Any set of terms cannot be expressed by explicit formula.
 To write the explicit formula, there should be a pattern that all terms follow.
 Apart from, the explicit formulas of arithmetic, geometric, and harmonic sequences, there can be any other formulas. For example, the explicit formula for the sequence 1, 4, 9, 16, 25, .... is a_{n} = n^{2} as every term is a square number of its position.
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Examples on Explicit Formulas

Example 1: What is the explicit formula for the sequence 3, 7, 11, 15, 19, 23, 27...?
Solution:
The given sequence is 3, 7, 11, 15, 19, 23, 27...
The given sequence represents an arithmetic sequence.
a = 3, and d = 7  3 = 4.
Explicit Formula is a_{n} = a + (n  1)d = 3 + (n  1)4 = 3 + 4n  4 = 4n  1
Answer: Therefore the explicit formula of the given sequence is a_{n} = 4n  1.

Example 2: What is the common difference of the sequence, if the explicit formula for the arithmetic sequence is a_{n} = 5n + 4?
Solution:
The given explicit formula is a_{n} = 5n + 4.
Let us find the 5^{th} and 6^{th }terms of the arithmetic sequence.
a_{5} = 5(5) + 4 = 29
a_{6} = 5(6) + 4 = 34
d = a_{6}  a_{5} = 34  29 = 5
Answer: Therefore the common difference of the arithmetic sequence is 5.
Note: The common difference is as same as the multiple of 'n' in the a_{n} which is 5.

Example 3: Find the explicit formula for the geometric sequence 4, 12, 36, 108....
Solution:
The given geometric sequence is 4, 12, 36, 108,.....
Here the first term is a = 4, and the common ratio is r = 12/4 = 3
a_{n} = ar^{n1}
a_{n} = 4.3^{n1}
Answer: Therefore the explicit formula for the geometric sequence is a_{n} = 4.3^{n1}.
FAQs on Explicit Formula
What is Explicit Formula In Algebra?
The explicit formula is useful to find any term of the sequence without the help of the previous terms of the sequence. The n^{th }term of the sequence forms the explicit formula and any term can be computed by substituting the value of n in the explicit formula. The explicit formula for the arithmetic sequence is a_{n} = a + (n  1)d, for the geometric sequence is a_{n} = ar^{n1}, and for the harmonic sequence is a_{n} = ar^{n1}.
How To Write Explicit Formula?
The explicit formula can be written from the n^{th} term of the sequence. For the arithmetic progression, a, a + d, a + 2d, a + 3d, .....a + (n  1)d, the n^{th} term forms the explicit formula of the sequence. And the explicit formula for this arithmetic sequence is a_{n} = a + (n  1)d. Similarly, the explicit formula can be computed for the geometric sequence and harmonic sequence.
What is Explicit Formula For Arithmetic Sequence?
The arithmetic sequence explicit formula can be easily computed from the term of the sequence. For the arithmetic sequence a, a + d, a + 2d, a + 3d, ........a + (n  1)d, and the n^{th }term of the sequence forms the explicit formula. Hence the explicit formula for the arithmetic sequence is a_{n} = a + (n  1)d.
What is Explicit Formula For Geometric Sequence?
The geometric sequence explicit formula is a_{n} = ar^{n1}. Here 'a' is the first term, and 'r' is the common ratio of the geometric sequence. Any term of the geometric progression can be computed using this explicit formula.
What is Explicit Formula For Harmonic Sequence?
The explicit formula for the harmonic sequence is a_{n} = 1/(a + (n  1)d). Here any term of the harmonic sequence can be computed using this explicit formula.
What is The Advantage Of Using Explicit Formula?
The explicit formula is useful to find any term of the sequence, without knowing the previous terms or any other term of the sequence. The explicit formula helps to find any term of the sequence with the least calculations.
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