1. An arithmetic progression (AP) is a sequence of numbers where the difference between successive terms is constant. This constant difference is called the common difference. 2. To define an AP, we need to know the first term (a1) and the common difference (d). The nth term of an AP can be calculated as an = a1 + (n-1)d. 3. Megha's annual salary increases by Rs. 1,000 each year, forming an AP. Her salary after 23 years can be calculated as a23 = Rs. 10,000 + 22(Rs. 1,000) = Rs. 32,000

2. • Patterns in daily life…..
Rungs
50 cm
38 cm
41 cm
44 cm
47 cm
29 cm
32 cm
35 cm
-3
-3
-3
-3
-3
-3
-3
Megha apllied for a job and got selected. She has been offered
A job with starting salary Rs 10,000/- with an annual increment
Rs 1,000/-. What will be the salary given to her for first five years?
Ans:
10,000……….11,000……..12,000………13,000……..14,000……..
15,000……

3. • These patterns in mathematics called as Progressions…….
• We have many progressions in mathematics like Arithmetic
progression,…….. Geometric progression……etc.

4. Arithmetic Progression (AP)
To understand AP lets take following examples,
• 1,2,3,4,………
• 200,150,100,50,……..
• -8,-6,-4,-2,0,………
• 3,3,3,3,……
Each of the number in the list is
called a TERM.
1 + 1 = 2
3 + 1 = 4
2 + 1 = 3
4 + 1 = 5
5
200 - 50 = 150
150 – 50 = 100
100 – 50 = 50
50 – 50 = 0
0
2
3

5. In these examples we observe, the pattern is formed by adding or
subtracting a fixed number to the preceding term. Such list of numbers
is said to form an Arithmetic progression.
So, an AP is, “a list of numbers in which each term is obtained by
adding a fixed number to the preceding term except the first term.”

6. Arithmetic Progression
• 51,53,55,57,……..71
• 200,150,100,50,……..
• 3,3,3,3,3,……..
The fixed number is
called Common
difference.
why we call it as
common difference?
𝟓𝟑
−𝟓𝟏
𝟐
𝟓𝟓
−𝟓𝟑
𝟐
𝟓𝟕
−𝟓𝟓
𝟐
Difference is 2
Common is 2
Therefore it is called as Common difference
Common difference is denoted by
‘d’
d=2
d= -50
d= 0
Common difference can be
Positive
Negative
Zero

7. Arithmetic Progression
• 51,53,55,57,……71
• 200,150,100,50,……….
Lets denote the
first term of AP
by a1
Second term by
a2
and nth Term by
an
Therefore
d = a2 – a1 = a3 – a2 =…….. = an – a(n-1)
a1, a2, a3, a4,………., an
Finite AP
Infinite AP
What is the minimum information you required to create an AP?
a1 = 3….. Can you find AP?
d = 2……. Can you find AP?
No
No
If you have both a1 = 3 and d = 2….. Can you find AP? Yes
To create an AP we need both first term and common difference..

8. Arithmetic progression
The general form of AP can be written as,
a + (a+d) + (a+2d) + (a+3d) +………….. + (a+ (n-1)d)
1st + 2nd + 3rd + 4th +………… + nth

9. Arithmetic Progression
Exercise
1) For the AP :
3
2
,
1
2
, −
1
2
, −
3
2
, … … . Write the first term a and the common
difference d.
Ans: Here, a =
3
2
, and d =
1
2
−
3
2
= −1
2) Write the first four terms of AP.
i) a=10, d=10
Ans : a1 = a = 10
a2 = a+d = 10+10 = 20
a3 = (a+2d) = (10+ 2 x 10) = 30
a4 = (a+3d) = (10 + 3 x 10) = 40

10. Arithmetic progression
nth dterm
Megha apllied for a job and got selected. She has been offered
A job with starting salary Rs 10,000/- with an annual increment
Rs 1,000/-. What will be the salary given to her for first five years?
Ans:
10,000……….11,000……..12,000………13,000……..14,000……..
15,000……
If I ask what will be the salary after 23 years … what you do…
If you add 1000 for every year the salary she get after 23
years is 32000
We can find this in a simple way…

11. • i.e., salary of 23rd year is = salary of 22nd year + 1000
= 10,000 + 1000 + 1000 + 1000 + … … + 1000 + 1000
21 times
= 10,000 + 22 x 1000
= 10,000 +(23-1) x 1000
= a + (n-1) x d
Therefore,
an = a + (n-1) d

12. Arithmetic Progression
• Find the 10th term of the AP : 2,7,12,…..
Ans : here a = 2, d = 5 and n = 10
We have, an = a + (n-1) d
a10 = 2 + (10-1) 5
a10 = 2 + 9 x 5
a10 = 2 + 45
A10 = 47

13. Homework…
• What is AP?
• What is least requirement we need to create AP?
• For following AP’s write first term and common difference
• 3,1,-1,-3
• -5,-1,3,7
• write the formula to find nth term of AP.
• Find the 25th term of AP: 10, 15,20