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Geometric Series

Last Updated : 01 May, 2024
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Geometric Series is a type of series where each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. Geometric series are characterized by their exponential growth or decay patterns. This growth or decay depends on whether the common ratio is greater than or less than one. In this article, we will discuss about geometric series in detail including formulas as well as derivation of the formula.

What is a Series?

A set of things that are in order is called a Sequence and when Sequences start to follow a certain pattern, they are known as Progressions. Progressions are of different types like Arithmetic Progression, Geometric Progressions, and Harmonic Progressions. 

The sum of a particular Sequence is called a Series.

A Series can be Infinite or Finite depending upon the Sequence, If a Sequence is Infinite, it will give an Infinite Series whereas, if a Sequence is finite, it will give Finite series.

Let’s take a finite Sequence:

a1, a2, a3, a4, a5, . . . an

The Series of this Sequence is given as:

a1 + a2 + a3 + a4 +a5 + . . . an

The Series is also denoted as :

\sum_{k=1}^{n}a_k

Series is represented using Sigma (∑) Notation in order to Indicate Summation.

What are Geometric Series?

In a Geometric Series, every next term is the multiplication of its Previous term by a certain constant and depending upon the value of the constant, the Series may be Increasing or decreasing.

Geometric Sequence is given as: 

a, ar, ar2, ar3, ar4,….. {Infinite Sequence}

a, ar, ar2, ar3, ar4, ……. arn {Finite Sequence}

Geometric Series for the above is written as:

a + ar + ar2 + ar3 + ar4 + . . . OR

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{Infinite Series}

a + ar + ar2 + ar3 + ar4 + . . . arn OR \sum_{k=1}^{n}a_k{Finite Series}

Where,

  • a = First term, and
  • r = Common Factor.

Geometric Series Formula

The Geometric Series formula for the Finite series is given as,

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\bold{{S_n =\frac{a(1-r^n)}{1-r}}}

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Where

  • Sn = sum up to nth term,
  • a = First term, and
  • r = common factor.

Derivation for Geometric Series Formula

Suppose a Geometric Series for n terms: 

Sn = a + ar + ar2 + ar3 + …. + arn-1 . . . (1)

Multiplying both sides by the common factor (r):

r Sn = ar + ar2 + ar3 + ar4 + … + arn . . . (2)

Subtracting Equation (1) from Equation (2):

(r Sn – Sn) = (ar + ar2 + ar3 + ar4 +. . . arn) – (a + ar + ar2 + ar3 + . . . + arn-1)

⇒ Sn (r-1) = arn – a

⇒ Sn (1 – r) = a (1-rn)

{S_n =\frac{a(1-r^n)}{1-r}}

Note: When the value of k starts from ‘m’, the formula will change.

\sum_{k=m}^{n}ar^k=\frac{a(r^m-r^{n+1}}{1-r}, when r≠0

For Infinite Geometric Series

n will tend to Infinity, n ⇢ ∞, Putting this in the generalized formula:

S_\infty = \sum_{n=1}^{\infty}ar^{n-1} = \frac{a}{1-r}; -1<{r}<1

nth term for the G.P. : an = arn-1

Product of the Geometric series

The Product of all the numbers present in the geometric progression gives us the overall product. It is very useful while calculating the Geometric mean of the entire series.

Geometric Mean

By definition, it is the nth root of Product of n numbers where ‘n’ denotes the number of terms present in the series. Geometric Mean differs from the Arithmetic Mean as the latter is obtained by adding all terms and dividing by ‘n’, while the former is obtained by doing the product and then taking the mean of all the terms.

Formula for Geometric Mean

Formula for Geometric Mean (GM) is given as:

\bar{X}_{geom}=\sqrt [n]{x_{1}.x_{2}.x_{3}...x_{n}}

Where,

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    is symbol for Geometric mean
  • x1, x2, x3, . . . , xn = terms present in the geometric series, and
  • n = number of terms present in the series.

Geometric Sequence Vs Series

Some of the common differences between Geometric Sequence and Series are listed in the following table:

AspectGeometric SequenceGeometric Series
DefinitionA sequence of numbers where each term is obtained by multiplying the previous term by a fixed, non-zero number (common ratio).The sum of terms in a geometric sequence.
General Forma, ar, ar2, ar3, ar4, . . .a + ar + ar2 + ar3 + ar4 + . . .
Example2, 6, 18, 54, . . .2 + 6 + 18 + 54 + . . .

Read More,

Sample Questions on Geometric Series

Question 1: What is the Geometric mean 2, 4, 8?

Solution: 

According to the formula, 

=\sqrt [3]{(2)(4)(8)}\\=4

Question 2: Find the first term and common factor in the following Geometric Progression:

4, 8, 16, 32, 64,….

Solution: 

Here, It is clear that the first term is 4, a=4

We obtain common Ratio by dividing 1st term from 2nd:

r = 8/4 = 2

Question 3: Find the 8th and the nth term for the G.P: 3, 9, 27, 81,….

Solution: 

Put n=8 for 8th term in the formula: arn-1

For the G.P : 3, 9, 27, 81….

First term (a) = 3

Common Ratio (r) = 9/3 = 3

8th term = 3(3)8-1 = 3(3)7 = 6561

Nth = 3(3)n-1 = 3(3)n(3)-1

= 3n

Question 4: For the G.P. : 2, 8, 32,…. which term will give the value 131073?

Solution: 

Assume that the value 131073 is the Nth term,

a = 2, r = 8/2 = 4

Nth term (an) = 2(4)n-1 = 131073

4n-1 = 131073/2 = 65536

4n-1 = 65536 = 48

Equating the Powers since the base is same:

n-1 = 8

n = 9

Question 5: Find the sum up to 5th and Nth term of the series: 1, \frac{1}{2},\frac{1}{4},\frac{1}{8}...

Solution: 

a= 1, r = 1/2

Sum of N terms for the G.P, {S_n =\frac{a(1-r^n)}{1-r}}                                              

 = \frac{1(1-(\frac{1}{2})^n)}{1-\frac{1}{2}}

 = 2 (1-(\frac{1}{2})^n)

Sum of first 5 terms ⇒ a52 ( 1-(\frac{1}{2})^5)

2 ( 1-(\frac{1}{32}))

(\frac{31}{16})

Question 6: Find the Sum of the Infinite G.P: 0.5, 1, 2, 4, 8, …

Solution:

Formula for the Sum of Infinite G.P: \frac{a}{1-r} ; r≠0

a = 0.5, r = 2

S= (0.5)/(1-2) = 0.5/(-1)= -0.5

Question 7: Find the sum of the Series: 5, 55, 555, 5555,… n terms

Answer

The given Series is not in G.P but it can easily be converted into a G.P with some simple modifications.

Taking 5 common from the series: 5 (1, 11, 111, 1111,… n terms)

Dividing and Multiplying with 9: \frac{5}{9}(9+ 99+ 999+...n terms)

⇒ \frac{5}{9}[((10+(10)^2+(10)^3+...n terms)-(1+1+1+...n terms)]

\frac{5}{9}[(\frac{10((10)^n-1)}{10-1})-(n)]

⇒ \frac{5}{9}[(\frac{10((10)^n-1)}{9})-(n)]

FAQs on Geometric Series

Define geometric series.

A geometric series is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio.

What are the two types of geometric series?

Geometric series can be classified into two types based on the behavior of their common ratio r:

  • Convergent Geometric Series (|r| < 1)
  • Divergent Geometric Series (|r| ≥ 1)

What is the general formula for the geometric series?

The general formula for the sum of a finite geometric series is:

{S_n =\frac{a(1-r^n)}{1-r}}

Can the values of ‘a’ and ‘r’ be 0?

No, the value of a≠0, if the first term becomes zero, the series will not continue. Similarly, r≠0.



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