Arithmetic and Geometric Progressions

All the concepts relating to arithmetic, geometric and harmonic progressions are listed and explained below.

What is Progression in Arithmetic and Geometric Progressions?

A sequence containing terms that follow a certain pattern is called progression.

Introduction to Arithmetic, Geometric and Harmonic Progressions

Progressions are also known as sequences and series. In progressions, a series of numbers are formed with an interlink between them. so, by predicting their order we can find the next number in series or missing number, the sum of the series, etc. They are arithmetic progression, Geometric progression, and Harmonic progression. Let’s see them in detail.

What is Arithmetic Progression?

Arithmetic Progression is a sequence of numbers in which the next term can be obtained by adding or subtracting a fixed number with the preceding term. It is represented by AP.
The general form of Arithmetic Progression is a, a+d, a+2d, a+3d…………….so on
Here
‘d’ is a fixed number and is obtained by the difference of any two consecutive terms in the arithmetic progression.
‘a’ is the first term of arithmetic progression.
If the number of terms in AP is ‘n’ then it can be represented as a, a+d, a+2d, ……..a+(n-1)d.
The last term in the series is given as

a_n = a₁+(n-1)d
here a_n is the last term in AP sequence
a₁ is the First term
d is the difference of consecutive terms in AP sequence.

The last term is also denoted by ‘l’.

l= a_n = a₁+(n-1)d

Example:
Find 7^th term of the AP sequence 3, 7, 11, 15………
Solution:
First-term in AP sequence is a=3
The difference between any two consecutive terms is d= 7-3 = 11-7 =15-11 = 4

i, e d=4
Term to find in sequence is n = 7
We have a_n = a₁+(n-1)d
substituting the values
 a₇=3+(7-1)×4
⇒ 3+6 × 4
⇒ 3+24
a₇=27 
Therefore the 7^th term of given AP sequence is 27.

Sum of Arithmetic Progression

For an arithmetic progression consisting of ‘n’ terms the first term as ‘a’ and common difference between terms is ‘d’ then, the sum of all the terms in AP is given by S.

The formula for the Sum of Arithmetic Progression

\begin{align*}
\begin{aligned}S=\dfrac {n}{2}\left[ 2a+\left( n-1\right) d\right] \\ Also \ written \ as \\
S=\dfrac {n}{2}\left[ a+a+\left( n-1\right) d\right] \\ S=\dfrac {n}{2}\left[ a+a_{n}\right] \end{aligned}
\end{align*}

Example for Sum of numbers in Arithmetic Progression

Find the sum of 10 terms of the AP sequence 5,4,3……
Solution:
First-term in AP sequence is a=5
The difference between any two consecutive terms is d= 4-5= -1.
Number of terms n=10

We have
\begin{align*} \begin{aligned}S=\dfrac {n}{2}\left[ 2a+\left( n-1\right) d\right] \\
=\dfrac {10}{2}\left[ a\times 5+\left( 10-1\right) \left( -1\right) \right] \\
=5\left[ 10-9\right] \\ =5 \\
Therefore \ sum \ of \ first \ 10 \ terms \ of \ AP \ is \ 5. \end{aligned}
\end{align*}

NOTE:
n^th term of AP is also represented as difference of sum of n terms and sum of first (n-1) terms of it.
\begin{align*}
a_{n}=S_{n}-S_{n-1}
\end{align*}

Arithmetic Mean

What is Arithmetic Mean?

Arithmetic Mean is the sum of all the given numbers in series divided by the total number of terms in the AP.
Example:

Let 2, 3 4 are in AP.

Arithmtic mean =(2+3+4) ⁄ 3
=9 ⁄ 3
=3
Here arithmetic mean is 3. This is the average of 2 & 4.
Therefore if a, b, c are three terms in AP then the middle term(b) is its Arithematic mean.

\begin{align*}
b=\dfrac {a+c}{2}
\end{align*}

What is a Geometric Progression?

Geometric progression is a sequence of numbers in which the next term can be obtained by multiplying a fixed number to the previous number.
This fixed number is called the common ratio and is represented by ‘r’.
It is obtained by dividing any two consecutive terms.
General Form of a geometric progression is a, ar, ar², ar³…………..ar^n-1
We have finite GP and Infinite GP.

• If the number of terms in GP series is finite then it is finite GP.
  a, ar, ar², ar³…………..ar^n-1
• If the number of terms in GP series is infinite then it is infinite GP.
  a, ar, ar², ar³…………..

Example of Geometric Progression

Let us consider a series of numbers that are in GP 2, 4, 8, 16, 32, 64
Here
First-term is a=2
The ratio of consecutive terms is r= 4 ⁄ 2= 8 ⁄ 4 =16 ⁄ 2 =2
i,e r=2

The generic term of Geometric Progression.

Let ‘r’ is the ratio of any two consecutive terms in the GP series.
Let ‘m’ is a term in this series it can be represented as

r_m=ar^m-1
r_m is m^th term
a is first term

If the GP consists of ‘n’ terms then n^th term or last term is given as 
\begin{align*}
r_{n}=ar^{n-1}
\end{align*}

Example:

Find the 10^th term in GP 2, 4, 8, 16, 32, 64

Here 
a=2, r=2, n=10
∴ r₁₀ = 2×2^10-1 =2×2⁹
r₁₀ =1024

The 10^th term of given GP is 1024.

Sum of Geometric Progression

For a Geometric progression consisting of ‘n’ terms with the first term ‘a’ and common ratio of consecutive terms in a sequence is ‘r’ then, sum of a geometric progression is given by S.

Sum of Geometric Progression Formula

\begin{align*}
\begin{aligned}S=\begin{cases}\dfrac {a\left( r^{n}-1\right) }{r-1} \ \ if \ r >1 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Here \ r\neq 1
\\ \dfrac {a\left( 1-r^{n}\right) }{1-r } \ \ if \ r <1\end{cases}
\end{aligned}
\end{align*}
For infinite Gemetric progression, sum of terms is given by
\begin{align*}
S=\dfrac {a}{1-r} \ \ \ \ \ Here \ r <1
\end{align*}

Example:

Find the sum of n^thterm of geometric progression 1, 1 ⁄ 4, 1 ⁄ 16 ……….

Here a=1. r= 1 ⁄ 4

\begin{align*}
\begin{aligned}
Sum \ of \ n \ terms \ =\dfrac {a\left( 1-r^{n}\right) }{1-r} \ \ r <1 \\ \\
=\dfrac {1\times \left[ 1-\left( \dfrac {1}{4}\right) ^{n}\right] }{1-\dfrac {1}{4}} \\ =4-\dfrac {1}{4^{n-1}}\end{aligned}
\end{align*}

Geometric Mean (GM)

Geometric mean is the root taken after muliplying the terms in geometric progression. Root depends on number of terms in sequence.

Example:

Let us consider geometric progression 1, 2, 4
Geometric mean = ³√1×2×4 = ³√8 =2
∴ Gm here is middle term 2 obtained by square rooting 1 & 4.
i, e √1×4 =2
Let a, b, c are in GP, then Geometric mean (b)=√ac

Till know we have clearly discussed Arithmetic and Geometric Progressions with definitions and their means. Now lets know about harmonic progression.

Harmonic Progression

Harmonic progression is a sequence of numbers whose reciprocal form a sequence of terms which are in AP.
i,e if a₁, a₂, a₃, ……are in HP then their reciprocals 1 ⁄ a₁ , 1 ⁄ a₂ , 1 ⁄ a₃ ………… are in AP.

To solve the HP we convert them to AP and then solve for answer.

General form of Harmonic Progression

Generally it is represented as

\begin{align*}
\dfrac {1}{a},\dfrac {1}{a+d},\dfrac {1}{a+2d}\ldots \ldots \dfrac {1}{a+\left( n-1\right) d}
\end{align*}

General n^th term of Harmonic Progression

\begin{align*}
It \ is \ represented \ by \ \dfrac {1}{a+\left( n-1\right) d}
\end{align*}

Relation among AM, GM and HM in Arithmetic and Geometric Progressions

Let us consider two numbers a & b their

\begin{align*}
\begin{aligned}AM=\dfrac {a+b}{2} \ \ \ \ \ \
GM=\sqrt {ab} \ \ \ \ \ \
HM=\dfrac {2ab}{a+b}\\ \\
Since, \ \ \ \ \ \dfrac {a+b}{2}\geq \sqrt {ab}\geq \dfrac {2ab}{a+b} \\ \\
Therefore \ \ \ \ \ \ \ \ \
AM\geq GM\geq HM\\ \\ \left( GM\right) ^{2}=AM\times HM\\ \\ ab=\dfrac {a+b}{2}\times \dfrac {2ab}{a+b} \\ \\ \therefore GM=\sqrt {AM\times HM}\end{aligned}
\end{align*}

Read More

Algebraic Expressions	Algebraic Properties
Algebra Multiplication Rules	Algebraic Equation List