Arithmetic Progression
Arithmetic Progression (A.P) is a sequence of numbers where difference between the two consecutive numbers is same.For example,
In an Arithmetic Progression: 1, 4, 7, 10, 13, .....
the difference between the all two consecutive numbers is same.
4 -1 = 3
7 - 4 = 3
10 - 7 = 3 and so on
Arithmetic Progression is also called Arithmetic Sequence.
nth term of an Arithmetic Progression
If a1 is the first term and d is the common difference, then nth term of an Arithmetic Progression can be found by using the formulathe second, third, fourth, ....., and nth term can be found as,
a2 = a1 + d
![](implies.png)
a3 = a1 + 2 d
![](implies.png)
a4 = a1 + 3 d
![](implies.png)
a5 = a1 + 4 d
![](implies.png)
.
.
.
a10 = a1 + 9 d
![](implies.png)
a11 = a1 + 10 d
![](implies.png)
a12 = a1 + 11 d
![](implies.png)
Practice Problems
Example 1. 3, 9, 15, 21, ........, a17 = ?Solution
Here, a1 = 3 and d = 6
a17 = a1 + 16 d
a17 = 3 + 16 ( 6 )
= 3 + 96
= 99 Answer
Example 2. If the 5th term of an A.P is 16 and the 20th term is 46. What is its 12th term ?
Solution
a5 = 16
![](implies.png)
![](implies.png)
a20 = 46
![](implies.png)
![](implies.png)
Subtract (1) from (2)
![](math-71.png)
d = 2
Put d = 2 in (1)
16 = a1 + 4 ( 2 )
16 = a1 + 8
a1 = 8
Now,
a12 = a1 + 11 ( d )
a12 = 8 + 11 ( 2 )
a12 = 8 + 22
a12 = 30 Answer
Example 3. Find the 13th term of the sequence x, 1, 2 - x, 3 - 2x
Solution
Here,
a1 = x
d = 1 - x [second term - first term]
a13 = a1 + 12 d
a13 = x + 12 ( 1 - x )
a13 = x + 12 - 12x
a13 = 12 - 11x Answer
Arithmetic Series
The sum of an Arithmetic Progression is called an Arithmetic Series.For example,
1, 3, 5, 7, ........., 99 is an Arithmetic Progression, and
1 + 3 + 5 + 7 + ..... + 99 is an Arithmetic Series.
Sum of an Arithmetic Sequence
Formula for finding Sum of an Arithmetic Progression isSolution
Here,
a = 2, n = 12 and d = 6 - 2 = 4
By using formula for sum of an arithmetic series
![](math-72.png)
Answer
A-Level Math November 2011 - 9709/11
Example 5. The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200. Find the seventh term.
Solution
Here,
![](math-73.png)
Subtract (4) from (3)
![](math-74.png)
Put d = 6 in (3)
![](math-75.png)
Now,
a7 = a + 6 d
= -7 + 6 ( 6 )
= -7 + 36
= 29 Answer