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Sequence

Sequence
Sequence

A succession of numbers each of which is formed according to the same fixed law is called a sequence. In sequence, there is a simple rule for going from one number to another.

Let us consider the following set of numbers,

  1. 1,3,5,7.....
  2. 2,4,6,8.....
  3. 5,6,9,11,17......

In (1) each number is greater than the preceding and by 2. Each number is formed by adding 2 to the preceding term. So, the number next to 7 is 7+ 2 = 9.

In (2) each number is double the preceding number. Each number is formed by multiplying the preceding number by 2. So, the number next to 16 is 16x2= 32.

But the set of numbers in (3) does not follow any rule. So, we cannot guess the number next to 17. Here, the set of numbers in (a) and (b) are arranged in order by some rule. So, these are the examples of sequences of numbers. But the set of numbers in (c) does not form a sequence.

Thus, an ordered list of the numbers connected by a rule is called a sequence of the numbers. This is the traditional definition of the sequence.

Types of sequence

Types of Sequence
Types of Sequence Source:www.slideshare.net

Generally, there are tow types of sequence. They are given below:

  • Finite sequence
    When the last term of tn is said to have a definite number in sequence then it is called finite sequence. For example, 1, 2, ,3 ,4 , 5, . . . . .9. Here the last term is 9.
  • Infinite sequence
    The term which has no last number or known term is said to be an infinite sequence. For example 2, 4, 6, 8, . . . . . . Here, the last term is unknown or infinite.

nth term of Linear Sequence

Each successive pair of terms having a common difference in a sequence is called a linear sequence.

nthterm of Quadratic Sequence

A quadratic sequence is a sequence of number in which there is same difference in the number.



  • The collection of numbers in which numbers are arranged in a certain rule is called a sequence.
  • When the last term of tn is said to have a definite number in sequence then it is called finite sequence.
  • The term which has no last number or known term is said to be an infinite sequence.
.

Questions and Answers

Click on the questions below to reveal the answers

Here,

Let the given sequences be a1, a2, a3, a4............

Now, a1=5

a2= 2=5-3

a3= -1=2-3

a4=-4=-1-3

Similarly the next term is a5= -4 -3 = -7.

Again from the given pattern

a1=5= 8-3 × 1

a2=2 =8-3× 2

a3= -1=8 - 3× 3

a4 = -4 =8-3× 4

⋅ ⋅ ⋅

⋅ ⋅ ⋅

⋅ ⋅ ⋅

an=8 - 3×n

∴ nth term (an) = 8 - 3n.

Now 12th term (a12) = 8 - 3× 12 = 8 - 36 = -28.Ans

 

Here,

Let the given sequences be a1, a2, a3, a4............

Now, a1 = 2

a2 = 4 =2 + 2

a3 = 6 = 4 + 2

a4 = 8 = 6 + 2

Similarly the next term is as = 8 + 2 =10.

Again from the given pattern.

a1=2=2 × 1

a2 = 4 = 2× 2

a3= 6 = 2× 3

a4 = 8 = 2× 4

⋅ ⋅ ⋅

⋅ ⋅ ⋅

⋅ ⋅ ⋅

an=2n

∴ nth term an= 2n

Again, 12th term a12 = 2 × 12

= 24 Ans.

soln: let the sequences be a1,a2,a3,a4,a6..........

nth term (an)=4n-3

∴ a1=4×1-3=4-3=1

a2=4×2-3=8-3=5

a3=4×3-3=12-3=8

a4=4×4-3=16-3=13

a5=4×5-3=20-3=17

a6=4×6-3=24-3=21

∴ The first 6 terms of the sequences are : 1,5,9,13,17 and 21 Ans.

soln: Let the sequences be a1,a2,a3,...........

nth term (an) = n2+2

∴ a1=12+2=1+2=3

a2=22+2=4+2=6

a3=32+2=9+2=11

a4=42+2=16+2=18

a5=52+2=25+2=27

a6=62+2=36+2=38

∴ The first 6 terms of the sequences are:3,6,11,18,27 and 38 Ans.

Here

let the sequence be a1,a2,a3........

nthterm (an)=(-2)n+1

a1 =(-2)1+1 =(-2)2 =4

a2=(-2)2+1=(-2)3=-8

a3=(-2)3+1=(-2)4=16

a4=(-2)4+1=(-2)5=-32

a5=(-2)5+1=(-2)6=64

a6=(-2)6+1=(-2)7=-128

∴ The first 6th term of the sequences are;

4,-8,16,-32,64 and -128 Ans.

Here given sequences,

an=2n

5 terms of the sequences by putting n =1,2,3,4, and 5 we get,

a1=2×1=2, a2=2 ×2 =4, a3=2×3=6,

a4 = 2 × 4 =8 and a5 =2×5=10

Sum of the 5 terms.

S5=a1+a2+a3+a4+a5

=2 + 4 + 6 + 8 + 10

=30 Ans.

Here, given

nth term of the sequences an=(-1)n 2n

Putting n=1,2,3,4 and5 we get,

a1=(-1)1.21=-2

a2=(-1)2.22=4

a3=(-1)3.23=-8

a4=(-1)4.24=16

a5=(-1)5.25=-32

Sum of the 5 terms

S5= a1+a2+a3+a4+a5

= -2+4+-8+16-32

= -42+20

=-22. Ans.

Here, given

nth term of the sequences an=(-1)n+1 2n

Putting n=1,2,3,4 and5 we get,

a1=(-1)1+1.21=(-1)2.2=2

a2=(-1)2+1.22=(-1)3 4=-4

a3=(-1)3+1.23=(-1)4.8=8

a4=(-1)4+1.24=(-1)5.16=-16

a5=(-1)5+1.25=(-1)6.32=32

Sum of the 5 terms

S5= a1+a2+a3+a4+a5

= 2-4+-8-16+32

= 42-20

=22. Ans.

Here,

Let the sequences be a1,a2,a3...........

nth term (an) =2n-1

∴ a1 = 2 × 1 -1 = 2 - 1= 1

a1 = 2 × 2 -1 = 4 - 1= 3

a2 = 2 × 3 -1 = 6 - 1= 5

a3 = 2 × 4 -1 = 8 - 1= 7

a4 = 2 × 5 -1 = 10 - 1= 9

a5 = 2 × 6 -1 = 12 - 1= 11

∴The first 6 terms of the sequences are:1,3,5,7,9 and 11. Ans.

0%
  • Find the U2 and U3 of a sequence if its nth term is defined as Un= 3Un-1-1 and U1=2

    5,14


    10,2


    4,12


    5,12


  • Find the sum of first four terms (S4):

    tn=4n+5

    33


    60


    57


    20


  • Find the sum of first four terms (S4):

    tn=3n+2

    78


    38


    22


    12


  • Find the sum of first four terms (S4):

    tn=5n+6

    56


    89


    74


    23


  • Find the sum of first four terms (S4):

    tn=6n-3

    48


    14


    39


    84


  • Find the sum of first four terms (S4):

    tn=2n

    30


    23


    46


    55


  • Find the sum of first four terms (S4):

    tn=n2

    20


    64


    30


    21


  • Find the sum of first four terms (S4):

    tn=n2+4

    36


    56


    46


    26


  • Find the sum of first four terms (S4):

    tn=n2-2

    38


    9


    11


    22


  • Find the sum of first four terms (S4):

    tn=(frac{2n}{n+1})

    8(frac{13}{20})


    6(frac{13}{20})


    5(frac{9}{22})


    5(frac{13}{20})


  • Find the sum of first four terms (S4):

    tn=(frac{2n-1}{2n+1})

    2(frac{564}{315})


    2(frac{134}{315})


    2(frac{134}{115})


    3(frac{194}{515})


  • Find the nth term the 3, 6, 9, 12, . . . . . . . . sequences

    2x
    3x
    5x
    4x
  • FInd the nth term of the quadratic sequences 1, 7, 17, 31, 49, . . . . . . . . . . 

    2x2 - 1
    2x3 - 3
    2x - 2
    3x2 - 3
  • Find the frist term of the sequences whose nth  term is tn  = n - 1

    1, 2, 3, 4, 5
    0, 1, 2, 3, 4
    0, 2, 4, 6, 8
    0, 1, 3, 5, 7
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