Arithmetic Sequence and Series

 

Motivation:

                         There exist a lot of sequences of numbers that can be seen in daily life. Some of the sequences are with some rules behind them; some of them are not with any regularity.   Here, we just study the most basic one.  Consider the sequence

 

                            1, 2, 3, 4, 5, 6, 7, …

 

                       What is the 100^th term of the sequence?  Furthermore, if

 

                                       S = 1 + 2 + 3 + 4 + 5 + … + 100 

 

                      Is there any way to find the sum instead of adding those terms one by one? Actually, there are many good ways to find the sum of those numbers. One of them is

 

                                       S = 1      + 2 +  3  + 4  + 5 + … + 100

                                       S = 100 + 99+ 98 + 97+96+ … + 1

                                    2S  = 101 + 101 + 101+ 101+ …    + 101

 

                 And there are 100 terms of 101 for 2S.  Thus,

 

                                      2S = 101  100 = 10100

 

                 So,      S = 5050 .

                 We will take a look at the more general cases.

 

Definition ( Arithmetic Sequence )

                    The following sequence is called as “arithmetic sequence” :

 

                             A, A+d, A+2d, A+3d, A+4d, …,  A+(n-1)d, A+nd, A+(n+1)d, …

 

                    The difference of successive terms is d. And “A” is the first term. Obviously, the nth term of this sequence is

                                                      A+(n-1)d

                    “d” is known as “common difference” of the arithmetic sequence.

 

Example:  For an arithmetic sequence, we know that the first term is 100 and the difference of successive is 4.  What is the 50^th term ?

        

           Sol:

                         100 + (50-1)  4 = 100+ 196 = 296

 

 

 

 

 

 

Definition ( Series )

                Given a sequence  a₁ , a₂ , a₃ , … ,  we can define a new sequence by

                                         S_n = a₁ + a₂ + a₃ + … + a_n  ,  n is an natural number .

                 So, the new sequence  is

                                         a₁ , a₁+a₂ , a₁+ a₂+ a₃ , …

                Then we can this new sequence as “series” .

 

Arithmetic Series

                  An arithmetic sequence is with first term A and common difference d.  Let’s use S_n to denote its sum of first n terms.  Then

 

                           S_n = A     +  (A+d)     +    (A+2d) +   (A+3d) + … + ( A+(n-1)d )

 

                 We write it again by aligning those terms in reverse order:

                           S_n =(A+(n-1)d) + (A+(n-2)d) + (A+(n-3)d)+ …..   + A

 

                 Adding up the two equalities, we have

 

                          2Sn = (2A+(n-1)d) + (2A+(n-1)d) + …. + (2A+(n-1)d)

 

                 There are n terms of  (2A+(n-1)d ).   So,

 

                                 2S_n =  n ( 2A+(n-1)d )

 

                       S_n =

 

 

Example:   Find the sum of the sequence

                                             2, 4, 6, 8, 10, 12, …., 100

 

         sol: 

                          This is an arithmetic sequence with first term 2 and common difference 2.

                     First, we find how many terms it has:

 

                                               (100-2) / 2 + 1 = 50

 

                           Thus, the sum is

                                              S = 50 ( 100 + 2 ) / 2 = 50  51 = 2550