Explicit Form Of A Sequence

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 - ane)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 u.s. check the explicit formulas and how to detect the explicit formulas of each of the sequence, with the assistance of examples, FAQs.

ane.	What are Explicit Formulas?
2.	How To Find Explicit Formulas?
3.	Explicit Formulas For Arithmetic Sequence
four.	Explicit Formulas For Geometric Sequence
5.	Explicit Formulas For Harmonic Sequence
half-dozen.	FAQs on Explicit Formulas

What are Explicit Formulas?

Explicit formulas are ever used to correspond whatever term of the sequence, without writing the other terms of the sequence. The significant of "explicit" is direct, something that can exist directly establish without knowing the other terms of the sequence. The terms of a sequence tin can be uniquely represented using a single formula, which is the explicit formula. The explicit formula helps to hands find any term of the sequence, without knowing its previous term. Hither are the explicit formulas of dissimilar sequences:

Arithmetics Sequence:a_n = a + (northward - one) d, where 'a' is the first term and 'd' is the common difference.
Geometric Sequence:a_northward = a r^{n - i}, where 'a' is the commencement term and 'r' is the mutual ratio.
Harmonic Sequence:a_n = i / [a + (n - one) d], where 'a' is the offset term and 'd' is the common difference of the arithmetics sequence formed past taking the reciprocals of the harmonic sequence.

Explicit Formulas

The to a higher place 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 corresponding explicit formulas.

Permit us consider a simple sequence of even numbers(2, four, 6, viii, .....) which can be uniquely represented using an explicit formula (a_n = 2n). Mostly, 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 Notice Explicit Formulas?

The explicit formula can exist computed from the n^th term of the given sequence. The northward^thterm of the arithmetics sequence is a_{due north} = 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 detect the explicit formula.

First, find the commencement term and the common difference of the arithmetic sequence.
Substitute the values in the north^th term of the explicit formula.
Simplify the n^th term mathematically to get the explicit formula.

The explicit formulas tin also exist 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 notice any term of the arithmetic progression.

Explicit formula for finding the north^thursday term of arithmetics sequence: a_north = a + (northward - 1)d

Let usa take an arithmetic sequence iv, vii, 10, thirteen, 16, ....... The get-go term of this sequence is a = four, and the common difference is d = seven - 4 = 3. Now we can easily discover 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 + (north - ane)d. The explicit formula is a_n = 4 + (n - 1)3 = 3n - 3 + iv, or we have a_north = 3n + 1.

Explicit Formula For Geometric Sequence

The geometric sequence explicit formula tin can be used to find whatsoever term of the geometric sequence. The explicit formula for the geometric sequence a, ar, ar^two, ar³, ........ar^{n - 1}is its north^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. Hither '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 = ar²/ar, and it is obtained by dividing a item term with its previous term.

Explicit formula for finding the due north^th term of geometric sequence: a_northward = ar^{n - 1}

Let us understand this past finding the explicit formula of a geometric sequence, 2, half dozen, 18, 54,... The first term of this sequence is a = ii, and the common ratio of this sequence is r = half-dozen/2 = 3. The explicit formula for the geometric sequence is a_northward = ar^{northward - ane} = 2.3^{n - 1}.

Explicit Formula For Harmonic Sequence

The harmonic sequence explicit formula is useful to easily detect any term of the harmonic sequence without finding the other terms of the sequence. The terms of the harmonic sequence are one/a, 1/(a + d), 1/(a + 2d), ane/(a + 3d), .....1/(a + (n - ane)d). Hither 1/(a + (north - 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_{due north} = i/(a + (northward - i)d)

This explicit formula of the harmonic sequence helps to easily observe any term of the sequence, without knowing the previous terms. For the harmonic sequence i/ii, 1/5, 1/eight, 1/11, .... the value of a = 2, and d = 5 - ii = 3, and the nth term of the sequence is 1/(ii + (northward - ane)3) = ane/(3n - i). Thus the harmonic sequence explicit formula for this sequence is a_n = ane/(3n - ane).

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 arithmetics, geometric, and harmonic sequences, there tin can be any other formulas. For example, the explicit formula for the sequence 1, 4, nine, sixteen, 25, .... is a_{due north} = due north² as every term is a foursquare number of its position.

☛ Related Topics:

Arithmetics Sequence Formula
Math Formulas
Algebra Formulas
Geometric Progression

Examples on Explicit Formulas

Example i: What is the explicit formula for the sequence 3, 7, 11, xv, 19, 23, 27...?

Solution:

The given sequence is 3, 7, 11, 15, nineteen, 23, 27...

The given sequence represents an arithmetics sequence.

a = 3, and d = seven - 3 = iv.

Explicit Formula is a_n = a + (n - 1)d = 3 + (northward - one)4 = 3 + 4n - iv = 4n - ane

Answer:Therefore the explicit formula of the given sequence is a_{due north} = 4n - ane.
Example two: What is the common deviation of the sequence, if the explicit formula for the arithmetic sequence is a_northward = 5n + iv?

Solution:

The given explicit formula is a_{due north} = 5n + 4.

Let us observe the 5^th and half dozen^thterms of the arithmetic sequence.

a₅ = 5(five) + iv = 29

a_half-dozen = 5(half-dozen) + iv = 34

d = a₆ - a₅ = 34 - 29 = five

Answer:Therefore the common difference of the arithmetics sequence is 5.

Annotation:The common difference is as same as the multiple of 'due north' in the a_northward which is 5.
Example 3: Find the explicit formula for the geometric sequence iv, 12, 36, 108....

Solution:

The given geometric sequence is four, 12, 36, 108,.....

Here the first term is a = four, and the common ratio is r = 12/iv = iii

a_n = ar^n-1

a_n = 4.3^north-1

Answer:Therefore the explicit formula for the geometric sequence is a_n = 4.3^n-one.

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Practice Questions on Explicit Formulas

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FAQs on Explicit Formula

What is Explicit Formula In Algebra?

The explicit formula is useful to observe any term of the sequence without the aid of the previous terms of the sequence. The n^thterm 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 arithmetics sequence is a_{due north} = a + (n - 1)d, for the geometric sequence is a_northward = ar^n-ane, and for the harmonic sequence is a_{due north} = ar^{due north-1}.

How To Write Explicit Formula?

The explicit formula can exist 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 arithmetics sequence explicit formula can be hands computed from the term of the sequence. For the arithmetics sequence a, a + d, a + 2nd, a + 3d, ........a + (due north - ane)d, and the north^thterm 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_northward = ar^north-1. Hither 'a' is the outset term, and 'r' is the common ratio of the geometric sequence. Any term of the geometric progression can exist computed using this explicit formula.

What is Explicit Formula For Harmonic Sequence?

The explicit formula for the harmonic sequence is a_n = 1/(a + (north - 1)d). Here whatever 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 whatsoever other term of the sequence. The explicit formula helps to observe whatever term of the sequence with the to the lowest degree calculations.