9.2 Arithmetic Sequences and Partial Sums

Arithmetic Sequences 

A sequence whose consecutive terms have a common difference 

A sequence is arithmetic if the differences between consecutive terms are the same. So, the sequence

is arithmetic if there is a number d such that

Example 1:

The sequence whose nth term is 4n + 3 is arithmetic. For this sequence, the common difference between consecutive terms is 4

7, 11, 15, 19, . . . , 4n + 3

  ^

11 - 7 = 4

The nth Term of an Arithmetic Sequence

The nth term of an arithmetic sequence has the form 

where d is the common difference between consecutive terms of the sequence and

An arithmetic sequence 

can be thought of as "counting by d's" after a shift of c units from d. For instance, the sequence

       2, 6, 10, 14, 18, . . .

has a common difference of 4, so you are counting by 4's after a shift of 2 units below 4 (beginning with 

 ).So, the nth term is 4n - 2. Similarly, the nth term of the sequence 

      6, 11, 16, 21, . . .

is 5n + 1 because you are counting by 5's after a shift of 1 unit above 5 (beginning with ).

Example 2:

Find the formula for the nth term of the arithmetic sequence whose common difference is 3 and whose first term is 2.

Solution

the formula for the nth term is . You know d = 3 and because , it follows that

So, the formula for the nth term is . The sequence therefore has the following form.

   2, 5, 8, 11, 14, . . . , 3n - 1, . . .

another way to find it is by writing the terms of the sequence.

        . . .

       2    2+3       5+3     8+3     11+3    14+3    17+3 . . .

       2       5          8         11       14       17         20    . . .

From these terms, you can reason that the nth term is of the form

  This is the graph -

The Sum of a Finite Arithmetic Sequence

There is a simple formula for the sum of a finite arithmetic sequence. A proof of the formula is given in Appendix A.

The Sum of a finite arithmetic sequence with n terms is 

 This formula works only for arithmetic sequences.

Example 3:

Find the sum: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 +19

Solution

To begin, notice that the sequence is arithmetic (with a common difference of 2). The sequence has 10 terms. So, the sum of the sequence is 

   = 50(101)


   = 5050