What is the Common Ratio of the Geometric Sequence Below

In Maths,
Geometric Progression
(GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term past a fixed number, which is chosen a common ratio. This progression is as well known as a
geometric sequence
of numbers that follow a design. Also, learn arithmetic progression hither. The mutual ratio multiplied here to each term to become the next term is a not-nil number. An case of a Geometric sequence is two, 4, eight, 16, 32, 64, …, where the common ratio is ii.

What is Geometric Sequence?

A geometric progression or a geometric sequence is the sequence, in which each term is varied by some other by a common ratio. The adjacent term of the sequence is produced when we multiply a constant (which is non-zero) to the preceding term. It is represented by:

a, ar, arⁱⁱ, ar^three, ar⁴, and so on.

Where a is the first term and r is the mutual ratio.

Notation:
It is to exist noted that when we divide whatsoever succeeding term from its preceding term, so we get the value equal to the common ratio.

Suppose we carve up the 3rd term by the 2nd term nosotros get:

arⁱⁱ/ar = r

In the same way:

ar³/ar²
= r

ar⁴/arⁱⁱⁱ
= r

Backdrop of Geometric Progression (GP)

Some of the of import backdrop of GP are listed below:

• Three non-zero terms a, b, c are in GP if and only if b^two
  
   = air-conditioning
• In a GP,
  
  
  
  Three consecutive terms can be taken as a/r, a, ar
  
  
  
  Four consecutive terms can be taken every bit a/r³
  
  , a/r, ar, ar³
  
  
  
  
  Five consecutive terms can be taken equally a/rⁱⁱ
  
  , a/r, a, ar, ar²
  
• In a finite GP, the product of the terms equidistant from the beginning and the stop is the same
  
  
  
  That means, t₁
  
  .t_n
  
   = t₂
  
  .t_n-1
  
   = t₃
  
  .t_northward-2
  
   = …..
• If each term of a GP is multiplied or divided past a not-cipher constant, then the resulting sequence is also a GP with the aforementioned common ratio
• The production and quotient of two GP’s is over again a GP
• If each term of a GP is raised to the power by the same non-zippo quantity, the resultant sequence is too a GP
• If a_i
  
  , a₂
  
  , a₃
  
  ,… is a GP of positive terms and then log a_i
  
  , log a_two
  
  , log a_three
  
  ,… is an AP (arithmetics progression) and vice versa

General Form of Geometric Progression

The general form of Geometric Progression is:

a, ar, ar², ar^three, ar⁴,…, ar^n-1

Where,

a = First term

r = common ratio

ar^n-1
= nth term

General Term or Nth Term of Geometric Progression

Allow a be the get-go term and r exist the common ratio for a Geometric Sequence.

Then, the 2d term, a₂
= a × r = ar

Third term, a₃
= a₂
× r = ar × r = ar^two

Similarly, nth term, a_n
= ar^n-1

Therefore, the formula to observe the nth term of GP is:

Note: The nth term is the last term of finite GP.

Common Ratio of GP

Consider the sequence a, ar, ar², ar³,……

First term = a

2d term = ar

3rd term = ar²

Similarly, nth term, t_north
= ar^{northward-ane}

Thus, the common ratio of geometric progression formula is given as:

Common ratio = (Any term) / (Preceding term)

= t_n
/ t_northward-i

= (ar^{n – one}
) /(ar^{n – 2})

= r

Thus, the full general term of a GP is given by ar^n-1
and the general form of a GP is a, ar, ar²,…..

For Example:
r = t₂
/ t₁
= ar / a = r

Sum of Northward term of GP

Suppose a, ar, ar², ar^three,……ar^n-1
is the given Geometric Progression.

Then the sum of north terms of GP is given past:

Southward_northward
= a + ar + arⁱⁱ+ arⁱⁱⁱ+…+ ar^{due north-1}

The formula to find the sum of n terms of GP is:

Where

a is the first term

r is the mutual ratio

north is the number of terms

As well, if the common ratio is equal to i, then the sum of the GP is given by:

Types of Geometric Progression

Geometric progression can be divided into two types based on the number of terms information technology has. They are:

• Finite geometric progression (Finite GP)
• Infinite geometric progression (Infinite GP)

These ii GPs are explained below with their representations and the formulas to find the sum.

Finite Geometric Progression

The terms of a finite One thousand.P. can be written equally a, ar, ar², ar³,……ar^n-ane

a, ar, ar², ar³,……ar^n-ane
is chosen finite geometric series.

The sum of finite Geometric series is given by:

Infinite Geometric Progression

Terms of an infinite Thou.P. can be written as a, ar, ar², arⁱⁱⁱ, ……ar^n-one,…….

a, ar, ar², ar³, ……ar^n-ane,……. is called infinite geometric series.

The sum of infinite geometric serial is given past:

\(\begin{assortment}{l}\sum_{k=0}^{\infty}\left(a r^{k}\right)=a\left(\frac{1}{1-r}\right)\end{assortment} \)

This is chosen the geometric progression formula of sum to infinity.

Geometric Progression Formulas

The listing of formulas related to GP is given below which will assistance in solving different types of bug.

• The general form of terms of a GP is a, ar, ar
  
  ²
  
  , ar
  
  ³
  
  , and so on. Hither, a is the first term and r is the mutual ratio.
• The nth term of a GP is T
  
  _north
  
  
  = ar^n-1
  
• Common ratio = r = T_n
  
  / T_n-ane
  
• The formula to summate the sum of the start northward terms of a GP is given by:
  
  
  
  
  S_n
  
   = a[(rⁿ
  
  – 1)/(r – 1)] if r ≠ 1and r > 1
  
  
  
  S_n
  
   = a[(1 – rⁿ
  
  )/(1 – r)] if r ≠ one and r < 1
• The nth term from the end of the GP with the final term 50 and common ratio r = fifty/ [r(n – 1)].
• The sum of infinite, i.e. the sum of a GP with infinite terms is S
  
  _∞
  
  = a/(1 – r) such that 0 < r < 1.
• If three quantities are in GP, then the eye one is called the
  
  geometric mean
  
  of the other two terms.
• If a, b and c are three quantities in GP, so and b is the geometric mean of a and c. This can be written every bit
  
  b²
  
   = ac
  
  or
  
  b =√ac
• Suppose a and r be the kickoff term and common ratio respectively of a finite GP with due north terms. Thus, the kth term from the end of the GP volition be = ar
  ^n-k
  .

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