5.4 Arithmetic sequences:

An arithmetic sequence is a sequence in which the difference between each consecutive term is constant. An arithmetic sequence can be defined by an explicit formula

a_n = d (n - 1) + a1

where d is the common difference between consecutive terms and a1 is the first term in the sequence.

Examples: For the sequence

1, 4, 7, 10, 13, 16…….

This sequence has a difference of 3 between each number. Therefore,

D = 3 and a1 = 1

Thus, the formula for the series is

a_n = 3(n – 1) + 1

= 3n – 3 + 1

a_n = 3n – 2

The sum of an infinite arithmetic sequence is either ∞ , if

d > 0 , or - ∞ , if d < 0 .

There are two ways to find the sum of a finite arithmetic sequence. In the first method, the value of the first term a1 should be known and the value of the last term an as well. Then, the sum of the first n terms of the arithmetic sequence is

S_n = n (a1 + a2)

2

To use the second method, you must know the value of the first term a 1 and the common difference d . Then, the sum of the first n terms of an arithmetic sequence is

S_n = na1 + n (dn - d)

2

Email Based Assignment Help in Arithmetic Sequences

To submit Arithmetic Sequences assignment click here.

Following are some of the topics in Sequences Of Numbers And Series in which we provide help: