In this presentation, you will learn sequences and arithmetic progression. How to find the terms, common differences, etc. I have given detailed solutions to each problem.
3. Sequence
A set of numbers where the numbers are arranged in a definite order, like the
natural numbers, is called a sequence.
โข Terms in Sequence
In a sequence, ordered terms are represented as ๐ก1, ๐ก2, ๐ก3 โฆ โฆ ๐ก๐
In general sequence is written as { ๐ก๐ }.
If the sequence is infinite, for every positive integer , there is a term ๐ก๐.
Example; 7, 14, 21, 28, 35โฆ.?
Here as ๐ก1 = 7, ๐ก2 = 14, ๐ก3 = 21, ๐ก4 = 28
4. Arithmetic Progression
A sequence in which the difference between any two consecutive terms is
constant then that sequence is known as arithmetic progression.
๐ก1 ๐ก2 ๐ก3 ๐ก4 ๐ก5 ๐ก6
d d d d d
d = ๐ก๐+1 โ ๐ก๐
In the general,
d = ๐ก2 โ ๐ก1 d = ๐ก3 โ ๐ก2 d = ๐ก4 โ ๐ก3
d= common difference
a= first term
6. 1. Which of the following sequences are A.P. ? If they are A.P. find the common
difference?
(1) 2, 4, 6, 8, . . .
Solution : From given sequence ,
๐ก1 = 2 , ๐ก2 = 4 , ๐ก3 = 6 , ๐ก4 = 8
โด ๐ก2 โ ๐ก1 = 4 โ 2 = 2
โด ๐ก3 โ ๐ก2 = 6 โ 4 = 2
๐๐ โ ๐๐ = ๐๐ โ ๐๐ = ๐๐ โ ๐๐
Since the difference between each term is common.
Therefore the given sequence is an A.P.
โด ๐๐จ๐ฆ๐ฆ๐จ๐ง ๐๐ข๐๐๐๐ซ๐๐ง๐๐ (๐) = ๐
โด ๐ก4 โ ๐ก3 = 8 โ 6 = 2
7. 2 . ๐ ,
๐
๐
, ๐ ,
๐
๐
โฆ โฆ . .
Solution: From given sequence ,
๐ก1 = 2 , ๐ก2 =
5
2
, ๐ก3 = 3 , ๐ก4 =
7
2
โด ๐ก2 โ ๐ก1 =
5
2
โ 2 =
1
2
โด ๐ก3 โ ๐ก2 = 3 โ
7
2
=
1
2
๐๐ โ ๐๐ = ๐๐ โ ๐๐ = ๐๐ โ ๐๐
Since the difference between each term
is common. Hence, the given sequence
is an A.P.
Common difference โด ๐ =
๐
๐
โด ๐ก4 โ ๐ก3 =
7
2
โ 3 =
1
2
8. (3) -10, -6, -2, 2, . . .
Solution : From given sequence ,
๐ก1 = โ10 , ๐ก2 = โ6 , ๐ก3 = โ2 , ๐ก4 = 2
โด ๐ก2 โ ๐ก1 = โ6 โ โ10 = 4
โด ๐ก3 โ ๐ก2 = โ2 โ (โ6) = 4
๐๐ โ ๐๐ = ๐๐ โ ๐๐
Since the difference between each term is common.
Hence, the given sequence is an A.P.
Common difference ๐ = ๐
9. (4) 0.3, 0.33, .0333, . . .
Solution : From given sequence ,
๐ก1 = 0.3 , ๐ก2 = 0.33 , ๐ก3 = 0.0333
โด ๐ก2 โ ๐ก1 = 0.33 โ 0.3 = 0.03
โด ๐ก3 โ ๐ก2 = 0.0333 โ 0.33 = โ0.2967
๐๐ โ ๐๐โ ๐๐ โ ๐๐
Since the difference between each term is not common.
Hence , the given sequence is not an A.P.
10. Solution : From given sequence ,
๐ก1 = 0 , ๐ก2 = โ4 , ๐ก3 = โ8 , ๐ก4 = โ12
โด ๐ก2 โ ๐ก1 = โ4 โ 0 = โ4
โด ๐ก3 โ ๐ก2 = โ8 โ โ4 = โ4
๐๐ โ ๐๐ = ๐๐ โ ๐๐ = ๐๐ โ ๐๐
Since the difference between each term is common.
Hence, the given sequence is an A.P.
Common difference (d) = -4
(5) 0, -4, -8, -12, . . .
โด ๐ก4 โ ๐ก3 = โ12 โ โ8 = โ4