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 formula
the second, third, fourth, ....., and nth term can be found as,
a2 = a1 + d
a2 = 1 + 3 = 4a3 = a1 + 2 d
a3 = 1 + 2 ( 3 ) = 7a4 = a1 + 3 d
a4 = 1 + 3 ( 3 ) = 10a5 = a1 + 4 d
a5 = 1 + 4 ( 3 ) = 13.
.
.
a10 = a1 + 9 d
a9 = 1 + 8 ( 3 ) = 25a11 = a1 + 10 d
a10 = 1 + 9 ( 3 ) = 28a12 = a1 + 11 d
a11 = 1 + 10 ( 3 ) = 31Practice 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
a5 = a1 + 4 d 16 = a1 + 4 d -------->(1)a20 = 46
a20 = a1 + 19 d 46 = a1 + 19 d -------->(2)Subtract (1) from (2)
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 is
Solution
Here,
a = 2, n = 12 and d = 6 - 2 = 4
By using formula for sum of an arithmetic series
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,
Subtract (4) from (3)
Put d = 6 in (3)
Now,
a7 = a + 6 d
= -7 + 6 ( 6 )
= -7 + 36
= 29 Answer