Arithmetic Geometric Sequence

Introduction

Arithmetic Geometric progression is a sequence of numbers in which each element is arranged in such an order that ratio of two consecutive terms always remains the same.

In mathematics, a sequence is a collection of numbers arranged in an order that each element in the sequence corresponds to the position and value of the next element. Sequence is the basis for calculating infinite series in advanced mathematics. In this tutorial, we will study arithmetic progression, nth term formula, sum of n terms of an arithmetic progression and some solved examples.

Sequence

A collection of numbers which can be arranged in an order so that the elements of the sequence can be repeated in any way. A sequence does not have a value, rather, each of its elements is defined by a formula, and can be calculated based on it. Progression can be of three types, namely, Arithmetic progression, Geometric progression, and Harmonic progression.

Arithmetic Sequence (AP)

Also known as an “arithmetic progression,” it is a sequence of numbers in which each element is arranged in such an order that each term in that sequence differs from the preceding element by the same value. Each element in the progression is called “term”.

For example: 2, 5, 8, 11, 14, 17, 20……..

Each term in the above sequence differs from its prior or next element by 3, which is called a “common difference” in an arithmetic progression. Common difference is the difference between consecutive numbers in an AP. We can also say that any element in the sequence can be calculated by either adding or subtracting a particular number from the element next to it.

Let there be a series,

x₁,x₂,x₃,x₄,x₅,...........x_n, where x₁ is the first term, and x_n is the last term.

And let “a” be the common difference in the above sequence.

Then,

$$\mathrm{x_2= x_1+a,}$$

$$\mathrm{x_3=x_2+a,}$$

$$\mathrm{x_4=x_3+a,}$$

$$\mathrm{\vdots}$$

$$\mathrm{x_n=x_{n-1}+a}$$

We can also write the sequence as,

General form of an AP can be written as

$$\mathrm{x_1,x_1+a,x_1+2a,x_1+3a,x_1+4a⋯,x_1+(n-1)a.}$$

Fig: The finite-sequence values are drawn on the graph in blue.

Geometric Sequence (GP)

A sequence of numbers in which each element is arranged in such an order that each term in that sequence can be calculated by either multiplying or dividing the number before or after it by a particular number.

For example: 3, 9, 27, 81, 243, 729……

Each term in the above sequence is a multiple of the number before it, which is called a “common factor” of the sequence; in this case 3.

Let us consider a series,

x₁,x₂,x₃,x₄,x₅,...........x_n, where x₁ is the first term, and x_n is the last term.

And let “m” be the common factor in the above sequence.

Then, we can also write the sequence as,

$$\mathrm{x_1,x_1 m,x_1 m^2,x_1 m^3,x_1 m^4\dotso,x_1 m^{n-1}.}$$

n^th term formula for a GP

nth term of a GP can be given by

$$\mathrm{x_n=x_1 r^{n-1}}$$

Sum of a GP, whose first term is x_1 and the common factor is r, is given by

$$\mathrm{S_n=\frac{x_1 (1-r^n)}{1-r}, where\: r≠1}$$

Solved Examples

1)Identify the following sequences and find the nth term of the sequence.

• 4,15,26,41,.......

• 4, 12, 36, 108……

Answer:

a. To identify the sequence we should first find the relation between the consecutive terms.

$$\mathrm{15-4=11}$$

$$\mathrm{26-15=11}$$

$$\mathrm{41-26=11}$$

The difference between each consecutive term is 11. Therefore, the sequence is an arithmetic progression.

The formula for finding the nth term of an AP is

$$\mathrm{nth term=x_1+(n-1)a}$$

$$\mathrm{x_1=4}$$

$$\mathrm{a=11}$$

Then the nth term of the sequence

$$\mathrm{=4+(n-1)11}$$

$$\mathrm{=11n-7}$$

Therefore, the nth term of the sequence is 11n-7.

b. To identify the sequence, we can divide each element by its preceding element other than the first term

$$\mathrm{12/4=3}$$

$$\mathrm{36/12=3}$$

$$\mathrm{108/36=3}$$

We can say that the above sequence is a geometric progression with the common factor 3. We know that nth term of a GP can be given by

$$\mathrm{\mathrm{x_n=x_1 r^{n-1}}}$$

Putting the values in the equation, we get

$$\mathrm{x_n=4×3^{n-1}}$$

$$\mathrm{x_n=4×3^n×3^{-1}}$$

$$\mathrm{x_n=4×3^n×\frac{1}{3}}$$

$$\mathrm{x_n=\frac{4}{3}×3^n}$$

Therefore, the nth term of the given sequence $\mathrm{x_n=\frac{4}{3}×3^n}$

2) Find the number of terms the following progressions.

• 1, 5, 9, 13, 17, ……….125

• 1, 2, 4, 8, 16, …………1024

Answer:

a. To find the number of terms in the above sequence, first we need to determine the type of progression

$$\mathrm{5-1=4}$$

$$\mathrm{9-5=4}$$

$$\mathrm{13-9=4}$$

The difference between each consecutive term is 4. Therefore, the sequence is an arithmetic progression.

The formula for finding the nth term of an AP is

$$\mathrm{nth term= x_1+(n-1)a}$$

$$\mathrm{ x_1=1}$$

$$\mathrm{a=4}$$

Then the nth term of the sequence

$$\mathrm{125=1+(n-1)4}$$

$$\mathrm{125 =4n-3}$$

$$\mathrm{4n=128}$$

$$\mathrm{n=32}$$

There are 32 terms in the given sequence.

b. To identify the sequence, we can divide each element by its preceding element other than the first term

$$\mathrm{2/1=2}$$

$$\mathrm{4/2=2}$$

$$\mathrm{8/4=2}$$

We can say that the above sequence is a geometric progression with the common factor 3. We know that nth term of a GP can be given by

$$\mathrm{x_n=x_1 r^{n-1}}$$

Putting the values in the equation, we get

$$\mathrm{1024=1×2^{n-1}}$$

$$\mathrm{1024=1×2^{n-1}}$$

We can write 1024 as 2¹⁰

$$\mathrm{2^{10}=2^{n-1}}$$

Because the base is same we can equate the powers

$$\mathrm{10=n-1}$$

$$\mathrm{n=11}$$

Therefore, the number terms in the given sequence 11.

Conclusion

Sequence is a collection of numbers which can be arranged in an order so that the elements of the sequence can be repeated in any way. Arithmetic sequence is a sequence of numbers in which each element is arranged in such an order that each term in that sequence differs from the preceding element by the same value. Geometric progression is a sequence of numbers in which each element is arranged in such an order that ratio of two consecutive terms always remains the same.

Differences between arithmetic and geometric progression.

Arithmetic progression	Geometric progression
Arithmetic progression is a sequence of numbers in which each element differs from the preceding element by the same value.	Geometric progression is a sequence where each term in the sequence is a multiple of the number before it.
An arithmetic progression is defined by a common difference between consecutive terms.	A geometric progression is defined by a constant factor between consecutive terms.
An AP is a linear sequence	A GP is an exponential sequence.
Example of an AP,	Example of a GP 10, 20, 40, 80,....
10, 30, 50, 70…..

FAQs

1. Name the types of progression in algebra.

Types of progression in algebra are −

• Arithmetic Progression (AP)

• Geometric Progression (GP)

• Harmonic Progression (HP)

2. What is a harmonic progression?

It is the sequence of the reciprocals of numbers which are an arithmetic progression.

3. How can we calculate an infinite sum with a common factor less than 1 ?

Geometric progression with an infinite number of terms and a common factor less than 1 can be calculated by the following formula −

$\mathrm{S=\frac{a}{1-r}}$, where “a” is the first term, “r” is the common factor.

4. What is the difference between a sequence and a series?

A sequence is an order of numbers based on defined properties. For example 2, 4, 6, 8….A series is a sum of numbers ordered in a sequence. For example 2+4+6+8…..

5. What is the relationship between AP and HP?

The reciprocal of the individual terms of an AP forms an HP.