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CBSE Class 10 Maths concepts for Chapter 5-Arithematic Progressions



Introduction:

We see lot of pattern in our daily lives.  For example, Rohan salary in his first year is 2000 and he will be getting increment of 500 every year,So his salary for First, second, third, fourth …. Will be of the form 2000,2500,3000,3500

So here we see the pattern in Rohan salary. The difference between two consecutive year salaries is constant. In the similar, world around us present us these pattern in many form. In this chapter, we will learn a special type of pattern called Arithmetic Progression

What is arithmetic progression? 

 An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant

Let us consider following series

(1)    1,5,9,13,17….

(2)    1,2,3,4,5,….

(3)    7,7,7,7….

All these sets follow certain rules. In first set 5 - 1 = 9 - 5 = 13 - 9 = 17 - 13 = 4

In second set 2 - 1 = 3 - 2 = 4 - 3  = 1

and so on.

Here the difference between any successive members is a constant

Such series are called Arithmetic Progression

Some Important points about AP

  1. The difference is called the common difference of the AP and It is denoted by d
  2. The members are called terms. The first member is called first term
  3. We can denote common difference  by d
  4. If a1, a2,a3,a4,a5  are the terms in AP then
    D=a2  -a1  =a3  - a2 =a4 – a3=a5 –a­4
  5. We can represent the general form of AP in the form
    a,a+d,a+2d,a+3d,a+4d………..
    Where a is first term and d is the common difference
  6. If the AP series has last term then it s finite Arithmetic Progression and if the AP series has infinite then it is called the Inifinite Arithmetic Progression

Example: 

Find if the below series is AP

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

Solution:

D=3=3=3=3

So it is AP

nth term of Arithmetic Progression

On the basis of above discussion we can consider the following series

a, a + d, a + 2d, a + 3d, .........................

Here a = 1, d = 4

a + d = 1 + 4 = 5

a + 2d = 1 + 2 x 4 = 9

and so on

Thus we can say that

a = First term

a + d = Second term

a + 2d = Third term

a + 3d = Fourth term and son on

nth term = a + (n - 1)d

Here First term = t1 = a

Second term = t2 = a + d

and hence, tn = a + (n - 1)d

d is called common difference and the series is called arithmetic progression

Sum of nth item in Arithmetic Progression

Let s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + + 8 + 9 + 10 and writting in reversed order

S = 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

Adding these two we get 2s = 11 + 11 + 11 + 11 + 11 + 11 + 11 = 11 + 11 + 11

= 10 X 11

s = (10 X 11)/2 = 55

In similar way, if

arithmetic progression

Adding we get

arithmetic progression


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