Understanding the Fundamentals of Arithmetic and Geometric Progression

In mathematics, a sequence is a list of objects (or events) that have been ordered sequentially, such that each member either comes before or after every other member. A series is a sum of a sequence of terms. That is, a series is a list of numbers with addition operations between them. This tutorial will teach you about Arithmetic and Geometric Progression, a part of sequence and series.

Sequence and series are important topics of mathematics. Several topics come under sequence and series. In this tutorial, you will learn in detail about Arithmetic and Geometric Progression. 

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What Is an Arithmetic Progression (A.P.)?

A sequence is called an arithmetic progression if the difference between a term and the previous term is always the same. 

an+1 - an  = constant (d) for all natural numbers, where an+1 is the term after an

The constant difference, generally denoted by d, is called the common difference.

Illustration 1: 

1, 5, 9, 13, … is an A.P. whose first term is 1, and the common difference is equal to 5 - 1 = 4. 

Arithmetic Progression Steps

Step 1: Obtain an

Step 2: Replace n by n+1 in an to get an+1

Step 3: Calculate an+1 - an

Step 4: If an+1 - an is independent of n, the given sequence is an Arithmetic Progression. Otherwise, it is not an Arithmetic Progression.

The following example will illustrate the procedure:

Example: Show that the sequence <an> defined by an = 4n + 5 is an A.P. Also, find its common difference. 

Solution: You have, an = 4n + 5

Replacing n by (n+1), we get

an+1 = 4(n+1) + 5 = 4n + 9

Now, an+1 - an = (4n + 9) - (4n + 5) = 4

Clearly, an+1 - an is independent of n and is equal to 4. So, the given sequence is an A.P. with a common difference of 4.

General Term of an Arithmetic Progression

Let “a” be the first term and “d” be the common difference of an A.P. Then, its nth term is given by:

an  = a + (n-1)d

where,

a = first term

d = common difference

Example 1: Show that the sequence 8,12, 16, 20, … is an A.P. Find the 16th term and the general term.

Solution: We have (12 - 8) = (16 - 12) = (20 - 16)  = 4. Therefore , the given sequence is an A.P. with common difference 4.

First term (a) = 8

16th term = a16 = a + (16 - 1)d = a +15d

a16 = 8 + 15*4 = 68

General Term = nth term = an = a + (n-1)d

an = 8 + (n-1)*4 = 4n + 4

Example 2: Which term of the sequence 72, 70, 68, 66, …. is 40?

Solution: Clearly, the given sequence is an A.P. with first term = 72 and 

common difference = - 2.

Let the nth term be 40. Then,

72 + (n-1)(-2) = 40

⇒ 72 - 2n + 2 = 40

⇒ n = 17

Sum of n Terms of an Arithmetic Progression

The sum (Sn) of n terms of an A.P. with the first term ‘a’ and common difference ‘d’ is 

Sn = n/2 [2a + (n-1)d]

Example: Find the Sum of 20 Terms of the A.P. 1, 4, 7, 10, ….

Solution: Let a be the first term and d be the common difference of the given A.P. 

Then a = 1, d = 3.

You have to find the sum of 20 terms of the given A.P. Putting a = 1, d = 3, n = 20 in Sn = n/2 [2a + (n-1)d], you get

S20 = 20/2 [2x1 + (20 - 1)x3] = 10 x 59 = 590

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Geometric Progression

A sequence of non-zero numbers is called a geometric progression (abbreviated as G.P.). If the ratio of a term and the term preceding it is always a constant quantity. 

The constant ratio is called the common ratio of the G.P.

General Term of a Geometric Progression

The nth term of a G.P. with first term a and common ratio r is given by 

an = arn-1

Example: Find the 9th Term and the General Term of the Progression 2, 6, 18, 54, …..

Solution: The given progression is clearly a G.P. with first term a = 2 and common ratio = 3.

9th term = a9 = ar(9 - 1) = 2*(3)8 = 13112

Sum of n Terms of a G.P.

The sum of n terms of a G.P. with first term ‘a’ and common ratio ‘r’ is given by

Sn = a[(rn-1)/(r-1)] if r ≠ 1

Example: Find the Sum of 7 Terms of the G.P. 3, 6, 12, ….

Solution: Here a  = 3, r = 2

S7 = a[(r7-1)/(r-1)] = 3[(27-1)/(2-1)] = 3(128 - 1) = 381

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Conclusion

With this, you have come to an end of this Arithmetic and Geometric Progression tutorial. You have learned how to calculate the nth term of any series and also the sum of the n terms in any given series.

Arithmetic and Geometric progressions are statistical fundamentals that are essential to working in Data Analytics. Learn all the essential concepts and perfect your skills in our Purdue University partnered Post Graduate Program in Data Analytics. Delivered in collaboration with IBM, you learn a curriculum with hand-picked skills that train you to become a world-class data analytics professional fast.

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