Sequences and Series - PDF

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					SEQUENCES AND SERIES
See the following sets of numbers, do they look cumbersome?

A A 1,5,9,13,17,21,25,29, '

     2   2   2      2       2   2   2   2
B A1 ,2 ,3 , 4 ,5 ,6 ,7 ,8 , '

     1 f ff 1 f 1 f 1 f 1 f 1 f
     ff 1f ff ff ff ff ff '
     ff ff ff ff ff ff ff
      f      f f f f f
     ff ff ff ff ff ff ff
CA     , ,   ,  ,  ,  ,   ,
     10 11 12 13 14 15 16

But look again closely. Do you see a specific order or pattern in each set? Isn’t it easy to
predict any term (say the 20th) in each of the above? We note that in A we add 4 each
time to get the next number, in B we take the square of the next integer and in C we take
the reciprocals of the next integer. Thus the 20th term in A is 77, in B is
                 1f
                 ff
                  ff
20 and in C is ff These are called sequences, where the subsequent terms are decided
   2
                   .
                39
by a given relation (which is expressed as a function f(n) ).

                                            SEQUENCE

A sequence is a function f whose domain is the set of all natural numbers (infinite
sequence) or some subset of natural numbers from 1 up to some larger number (finite
sequence). The values f(1), f(2), f(3). . . . are the terms of the sequence.

Let the terms of a sequence be given by the function by f n = n 2 + 1 where n is a natural number A
                                                                ` a

             f 1 = 1 + 1 = 2,                f 2 = 2 + 1 = 5,         f 3 = 3 + 1 = 10,
              ` a       2                     ` a     2                ` a   2
Here
          f 4 = 4 + 1 = 17,          f 5 = 1 + 1 = 26, ……A
             ` a        2                    ` a    2


The sequence becomes     2, 5, 10, 17, 26, ………………A


1, 5, 9, 13, 17 is a finite sequence with 5 terms.
1, 5, 9, 13, 17, 21,……… is an infinite sequence.

A sequence is represented by its range. f n = an is used to denote the range elements of
                                                ` a

the function. a1 , a2 , a3 , ……an , …A are the first term, second term, third term,….nth
term,…. respectively of the sequence.
Example : Give first 3 terms of the sequence an = n 2 + n + 1 .

Solution : Putting n = 1,2,3 in an = n 2 + n + 1, we get
        a1 =1 + 1 + 1 = 3,        a2 = 2 + 2 + 1 = 7,    a3 = 3 + 3 + 1 = 13
              2                          2                     2


        The sequence would be written 3, 7, 13,………
                                                         n
                                                       fffff
                                                       ffff
Example : Give first 3 terms of the sequence an = fffff.ffff
                                                      n +1
                                                       2
                                        n
                                      fffff
                                       ffff
                                       ffff
                                      fffff
Solution : Putting n = 1,2,3 in an = 2     , we get
                                     n +1
                1
              ffff 1f              2
                                 fffff 2f             3
                                                    fffff 3f
              ffff f
               ffff              ffff f
                                  ffff
        a1 = fffff f, a2 = fffff f, a3 = fffff ff
                    =                  =            ffff ff
                                                     ffff ff
                                                           =
            1 +1 3              2 +1 5            3 + 1 10
              2                  2                  2



                                        1f 2f 3f
                                         f f ffff
       The sequence would be written as f, f, ff, AAAAAAA
                                        3 5 10

                                       SERIES
If a1 ,a2 ,a3 ,a4 , ……AA ,an , ……… is a sequence, then the expression of their summation
a1 + a2 + a3 + a4 + ……A + an + …… is a series.

If we take m terms of a finite sequence, the series can be written with summation notation
 Σ (Sigma notation ).
  m
X ak     = a1 + a2 + a3 + a4 + ……A + am
 k=1
         = The sum of all , the value of k being all the natural numbers from 1 to k,
           k is called the summation variable A
                                         4
                                                1
Example 3 : Write in expanded form : X fffff  ffff
                                               ffff
                                             fffff
                                        k=1 k + 1
                                              3

                           1
                         fffff
                          ffff
                          ffff
                         fffff
Solution : Replace k in 3      with integers from 1 to 4 and add them
                        k +1
  4
         1           1       1       1         1          f f 1f 1f
                                                         1f 1f ff ff
       fffff
       ffff
        ffff
 X fffff = fffff fffff fffff fffff =
                   ffff fffff fffff fffff
                   ffff ffff
                    ffff ffff ffff
                                + 3ffff+ ffff
                        + 3                  ffff
                                                           + + ff ff
                                                          f f ff ff
                                                                   +
 k=1 k + 1       1 +1 2 +1 3 +1 4 +1
       3           3                         3
                                                         2 9 28 65
                                      SERIES IDENTITIES:
    n       n               n                                  n        n         n
   X ak + X bk = X ak + bk                                 X ak @ X bk = X ak @ bk
                                 b       c                                            b         c

   k=1     k=1          k=1                                k=1         k=1     k=1
    n               n                                       n
   X cak = c X ak                                          X c = cn
   k=1           k=1                                       k=1

                                                                nfff+fff2nffff
                                                                   n f 1 ff + 1
    n
            `           a                                   n
                                                                        `    a`            a
           n@1
        nfffffff
          fffffff
          ffffff
   X k = fffffff                                                 fffffffffffff
                                                                  fffffffffffff
                                                          X k = ffffffffffffff
                                                                   2

   k=1
            2                                             k=1
                                                                        6
                                                                          n + 1 2n + 1 3n 2 + 3n @ 1
                                                                                           ab       c
            n+1
    n     2
                `           a2                             n
                                                                        `    a`
                                                                       nfffffffffffffffffffffffff
         nffffffff
          ffffffff
           fffffff
   X k = ffffffff                                         Xk =           ffffffffffffffffffffffff
                                                                        fffffffffffffffffffffffff
                                                                        fffffffffffffffffffffffff
       3                                                           4

   k=1
             4                                            k=1
                                                                                          30


                    ARITHMATIC SEQUENCE & SERIES
A sequence where we start with a number a and repeatedly add a fixed constant d to get
the subsequent numbers, is called an Arithmatic Sequence.

If we start with 2 and add a fixed constant 3 to it repeatedly, we get the sequence
2, 5, 8, 11, 14, 17, AAAAAAAAAAAA This is an arithmatic sequence.

An Arithmatic Sequence is always of the form
                                 a, a + 2d, a + 3d, a + 4d, a + 5d, AAAAAAAAAAA
where we start with the number a and keep on adding d to get the subsequent terms.
The number a is the first term (can be written as a1 too) and d is the common
difference.

To get d in a given arithmetic sequence, subtract any term from its next term.
 d = a2 @ a1 = a3 @ a2 = a4 @ a3 = ……AA = an @ an @ 1 = ……A

        term of an arithmetic sequence is given by
The nth `
an = a + n @ 1 d     where an = nth term, a = a1 = first term, d = common difference A
              a




If S n is the sum of n terms of a finite arithmetic sequence, then
       nf       a nf
        ff          ff
 S n = f a + a n = f 2a + n @ 1 d
         `           B    `      a C
       2           2

Example : Write the first 5 terms of the arithmetic sequence 11,8,….. and find its 100th
term.

Solution : Here a1 = a = 11 and a2 = 8 A So d = a2 @ a1 = 8 @ 11 = @ 3 A
       Thus each term can be found by adding -3 to the previous term.
        Hence the first 6 terms are 11, 8, 5, 2, -1, -4.
        For the 100th term n = 100, and we have already found a = 11, d = @ 3
                    an = a + n @ 1 d
                             `      a
        Using
                    a100 = 11 + 100 @ 1 @ 3
                               `       a`   a

                         = 11 + 99 @ 3
                                   `  a

                         = 11 @ 297
                         = @ 286

Example : Find the sum of first 12 terms of the arithmetic sequence given by
an = 2n + 5 A

Solution : Putting n=1 and 2 we get,
a = a1 = 2 B 1 + 5 = 7, a2 = 2 B 2 + 5 = 9 A   So d = a2 @ a1 = 9 @ 7 = 2 A
                                    nf
                                     f
                                     ff
Using the summation formula S n = 2a + n @ 1 d
                                      B      ` a C
                                    2
              12f
               ff
                f
                f
               ff
we get S 12 =     2 B7 + 12 @ 1 2 = 216 A
                 B      `      a C
               2

                 GEOMETRIC SEQUENCE & SERIES :
A sequence where we multiply each term with a non-zero constant to get the next term, is
a Geometric Sequence.

If we start with 2 and multiply it repeatedly by a non-zero fixed number 3, we get the
                       2       3
sequence 2, 2 B3, 2 B3 , 2 B3 , AAAAAAAAAAAA or 2, 6, 18, 54, AAAAAAA This is a geometric
sequence.

A Geometric Sequence is always of the form
                            a, ar 2 , ar 3 , ar 4 , ar 5 AAAAAAAAAAAAAA
 where we start with the number a and keep on multiplying with r to get the subsequent
terms.
The number a is the first term (can be written as a1 too) and r is the common ratio.

r is the ratio of any term to its previous term,
     aff aff aff             anff
      f 3
 r = ff ff ff ……AA = ffff ……A
      2f f 4
        = f= f=  f            f
                             ffff
                              fff
                                   =
     a1 a2 a3               an @ 1

Examples of Geometric sequence :
       1f 1f ff ff
        f f 1f 1f
        f f ff ff                                             1f     1f
          , , ff ffAAAAAAAAA
                  , ,                                          f
                                                          a = f, r = f f
       2 6 18 54                                              2      3
       1, @ 10, 20, @ 40, 80, AAAAAAAAAA                  a = 1, r = @ 5
       2, 0.2, 0.02, 0.002, AAAAAAAAAA                    a = 2, r = 0.1
The nth term of a Geometric Sequence is given by
               an = a r n @ 1 where an = nth term, a = a1 = first term, r = common ratio A

A Geometric Series is the sum of the terms as indicated above. The sum of n terms of it
                      @rn
                    1fffff
                     fffff
                      fffff
is given by S n = a fffffwith r ≠ 1
                     1@r

Example : Write the 6th term and sum of first 6 terms of the geometric sequence
                     9ff
                4, 6, f, AAAAAAAAA
                     2
Solution :
                                                                          aff 6f 3f
                                                                            f f f
                                                                            f f
                                                                           ff f f
Here in this geometric sequence, a = 4, and a2 = 6, the common ratio r = 2 = = . So,
                                                                          a1 4 2
                                                            f g6 @ 1
                                                             3f
                                                              f
                                                              f         243f
 using an = a r   n@1
                        ,          6th term = a6 = 4 B                 = fff
                                                                          ff
   
				
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