Arithmetic Progression

Arithmetic Progression (AP)

If you look around, you will see various things that follow a pattern. The dates on the calendar, your roll numbers, the compound interest your bank calculates etc. All these things can be said to follow a sequence or a series. Here we are going to learn about one type of such sequence, called Arithmetic Sequence or Arithmetic Progression.

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What is Arithmetic Progression?

Arithmetic Progression is a sequence in which the difference between one term and the next is constant. This difference is called the common difference and is denoted by a “d”. 

Example:

Look at this series: 2,4,6,8,10,12,14,16,18….

As you can see, these are all even numbers. But did you know this is also an Arithmetic Progression? See the difference between two adjacent numbers. For every instance, the difference is 2, which is called the common difference. Therefore, this is an arithmetic progression.

Finite Arithmetic Sequence

If something is finite then it has a limit, an ending. So if the number of terms in an Arithmetic Progression has a limit, then the sequence is called a finite sequence, and the AP is called a Finite AP.

Example : 2,4,6,8

This sequence has 4 numbers, it’s a Finite AP.

Infinite Arithmetic Sequence

When the number of terms in a sequence is no limit, it goes on, then such a sequence is an infinite sequence and the AP is called an Infinite AP.

Example : 1,3,5,7,9,11,13,15…….

As you can see, the sequence does not have a limited number of terms, hence it’s an Infinite AP.

Download NCERT Solutions for Class 10 Mathematics

nth Term of an Arithmetic Progression

Here we are going to learn how to derive the nth term of an Arithmetic Progression whose first term is a1 and the common difference is d.

Now let us look at the following Arithmetic progression

a1, a2, a3, a4, a5, a6………..an

So here we will notice that to arrive at the next term we add the common difference.

a1 = a1

a2 = a1+d

a3 = a1+2d

a10 = a1+ 9d

an = a1 + (n-1)d

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Arithmetic Progression

Sum of first n terms of an Arithmetic Progression

Now let us derive the formula to calculate the sum of n terms in an AP.

As we already know, nth term of an AP is a1 + (n-1)d. So the sum, represented by “S” can be derived as follows

S = a1 + a2 + a3…… + an

S = a1 + (a1+d) + (a1+2d) + …… a1 + (n-1)d                                                                                 …(I)

Now we write the terms in reverse order to arrive at S

S = a1 + (n-1)d + a1 + (n-2)d + a1 + (n-3)d…….+ a1                                                                                 …(II)

Adding equation (I) and (II), we get

2S = [2a+(n-1)d] + [2a+(n-1)d] + ……. [2a+(n-1)d] (n terms)

2S = n [2a+(n-1)d]

S = n/2 [2a+(n-1)d]

Solved Example for You

Question 1: Find the sum of all integers between 1 and 100, that are divisible by 2 or 5.

Answer : So here we will find the sum of all integers from 1 to 100 that are divisible by 2 or 5

S = (Sum of all integers divisible by 2) + (Sum of all integers divisible by 5) – (Sum of all integers divisible by 2 and 5 )

The AP of all numbers divisible by 2 is 2,4,6,8…. 100.

n = (an-a)/d + 1

n = (100 – 2)/2 + 1

n = 50

S = n/2 [2a+(n-1)d]

S = 50/2 [102]

S=  2550                                                                                                              …(I)

The AP  of all the numbers divisible by 5 is 5,10,15….100

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

n = [(100 – 5)/5] + 1

n = 20

S = n/2 [2a+(n-1)d]

S = 20/2 [105]

S = 1050                                                                                                             …(II)

The AP of all numbers divisible by both 2 and 5 is 10,20….100

n = 10

S = n/2 [2a+(n-1)d]

S = 10/2 [110]

S = 550                                                                                                                 …(III)

Hence the required sum = 2550 + 1050 – 550 = 3050

Question 2: What is the formula for the arithmetic sequence?

Answer: It is a sequence where the dissimilarity between each consecutive term is constant. The arithmetic sequence is definable by an obvious formula in which ‘a = d (n – 1) + c’, Here, d is the mutual difference between the consecutive terms, and here, c = ‘a1’.

Question 3: Define the arithmetic sequence?

Answer: Mathematically, an arithmetic progression i.e. AP is a sequence of the numbersjust like the difference between these consecutive terms is constant. For example, this sequence ‘5, 7, 9, 11, 13, 15, 17, 19’ is an AP (arithmetic progression) with a common difference of ‘2’ (d = 2).

Question 4: What is an arithmetic pattern?

Answer: The arithmetic pattern is one of the easiest series to study. It contains adding or subtracting from a common difference (d), for generating a string of the numbers that are interrelated to each other.

Question 5: Tell the difference between the arithmetic sequence and the geometric sequence.

Answer: The arithmetic sequence is a collection of numbers that go according to a pattern based upon one particular rule. Whereas, the geometric sequence is basically a list of the numbers that are divided or multiplied by a similar amount.

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