Arithmetic Progression and Geometric Progression

Arithmetic Progression and Geometric Progression – Definition, Formulas, and solved examples

In Maths, there are three different progressions namely Arithmetic Progression, Geometric Progression, and Harmonic Progression. Each of these progressions follows a set pattern in a series or sequence. For example, consider a series with the terms 1,3,5,7,9,11, and so on. We can see that each term increases by a value of 2. Consider an example of a population tree below. Notice how each layer increases by a multiple of 2.

These are some examples of progressions. Let us learn Arithmetic and Geometric progressions in detail here.

Definition of AP and GP:

Consider the series having the terms 1, 6, 11, 16, 21, and so on. Notice how the difference between any two adjacent terms is the same. The difference between 6 and 1 or 11 and 6 or 16 and 21 is always 5 and is constant. Such a series in which the common difference between any two successive terms is constant is known as an arithmetic progression.

An Arithmetic Progression that has an infinite number of terms is known as an infinite arithmetic progression.

Now, consider a series containing the terms 9, 18, 36, 72, and so on. The ratio of every two successive terms in this series i.e., \frac{18}{9}=\frac{36}{18}=\frac{72}{36}=2\text{     }(\frac{T_2}{T_1}=\frac{T_3}{T_2}=\frac{T_4}{T_3}). Similarly, consider a series having 270, 90, 30, 10, and so on. The ratio of every two consecutive terms in this series is \frac{1}{3}.  A series in which the ratio of any two successive numbers is constant is known as geometric progression.

A Geometric Progression that has an infinite number of terms is known as an infinite geometric progression.

General Expression and the terms in AP and GP

Now that we have understood what AP and GP mean, let us learn the various terms in an AP and GP and their general form.

We know that the difference between any two consecutive numbers in an AP is always the same. So, to know the value of the next term in an AP, we have to add the common difference to the previous term in that AP. So, the general form of an arithmetic progression is a,a+d,a+2d,a+3d,......,a+(n-1)d.

Where,

a = first term

d = common difference

n = number of terms in the Arithmetic Progression

Similarly, we know that the ratio of any two consecutive numbers in a GP is always constant. So, to know the value of the next term in a GP, we have to multiply the previous term with the common ratio. So, the general form of a Geometric Progression is a,ar,ar^2,ar^3,......,ar^{n-1}.

Where,

a = first term

r= common ratio

n = number of terms in the Geometric Progression

Formulas in AP and GP

nth term of an AP Series

In an arithmetic progression where the first term and the common difference is known, the nth term of the series can be calculated using the formula T_n=a+(n-1)d.

Where,

a = first term

d = common difference

n = number of terms in the Arithmetic Progression

Number of terms in an AP

When the first and last terms of an AP and the common difference is known, then we can calculate the number of terms in the arithmetic progression using the formula n=\frac{l-a}{d}+1.

Where,

a = first term

l = last term

d = common difference

Sum of first n terms in an AP

When we know the first term and the common difference in the AP, the formula for calculating the sum of an arithmetic progression is S=\frac{n}{2}[2a+(n-1)d].

Where,

a = first term

d = common difference

n = number of terms in the Arithmetic Progression

AP Formula summary

n^th term of a Geometric Progression:

We can calculate the n^th term of a GP using the formula T_n=ar^{n-1}.

Where,

a = first term

r = common ratio and

n = number of terms in the GP

To calculate the n^th term from the end of a GP where the last term is known, we can use the formula T_n=\frac{l}{r^{n-1}}.

Where,

l = last term

r = common ratio

n = number of terms from the end of the GP.

Sum of the first n terms of a geometric progression

If r=1,\text{ then } S_n=a+a(1)+a(1)^2+a(1)^3+....+a(1)^{n-1}=na.

If r>1,\text{ then } S_n=\frac{a(r^n-1)}{r-1}.

And when r<1,\text{ then } S_n=\frac{a(1-r^n)}{1-r}.

Where,

a = first term

r = common ratio and

n = number of terms in the GP

Sum of an infinite geometric progression

The sum of an infinite geometric series can be calculated using the formula

S=\frac{a}{1-r}, where r ≠ 0 and | r | < 1.

Formula summary

Difference between AP and GP

Solved Examples

Example: Identify the series which are in an AP and GP from below:

a. 1, 3, 7, 11, 15

b. 2, 11, 20, 29, 38

c. 4, 16, 64, 256

Solution:

a. The difference between T₂ and T₁ is 2 while the difference between T₃ and T₂ is 4. Since this series does not have a common difference, it is not an AP. Similarly, the ratio of T ₂ and T₁ is 3 while the ratio of T₃ and T₂ is 2.33. Since the ratio of two consecutive terms is not constant, it is not a GP.

b. The difference between T₂ and T₁, T₃ and T₂ is 9 and it is constant. So, it is an arithmetic progression series.

c. The ratio of T ₂ and T₁ is 4 and the ratio of T₃ and T₂ is also 4. We can see the ratio of all consecutive terms in this series is constant. So, this is a geometric progression series.

Example: Find the next term in each of the following series:

a. 4, 10, 16, 22, ___

b. 7, 21, 63, 189, ___

Solution:

a. From the given question, we can see that this is an arithmetic progression series with a common difference d = ( 10 – 4 = 16 – 10 = 22 – 16) 6.

To calculate the next term in the series, we have to add the common difference to the previous term. So the next term is 22 + 6 = 28.

b. From the given question, we can see that this is a geometric progression series with a common ratio r = 3.

To calculate the next term in the series, we have to multiply the common ratio with the previous term. So the next term is 189 x 3 = 567.