Sequence $ a_n$ is called the arithmetic sequence if every member of that sequence is equal to the previous member increased by a constant d in such way that:
$ a_{n + 1} = a_n + d$
Number $d$ is called the difference of arithmetic sequence.
Arithmetic sequence is uniquely identified by its first member and the difference $d$.
Why does arithmetic sequence have that name? Every member of arithmetic sequence, except for the first is arithmetic middle of two adjacent members.
$ a_n = \frac{a_{n + 1} + a_{n – 1}}{2}$
Example 5. Write first five members of arithmetic sequence whose first member is $3$ and whose difference is $4$.
$3, 7, 11, 15, 19$
Here you can also test previous statement:
$ a_2 = \frac{3 + 11}{2} = \frac{14}{2} = 7$
$ a_3 = \frac{7 + 15}{2} = \frac{22}{2} = 11$
$ a_4 = \frac{11 + 19}{2} = \frac{30}{2} = 15$
$ a_5 = \frac{15 + 23}{2} = \frac{38}{2} = 19$
Arithmetic sequence has few divisions, depending on the difference.
If $ d > 0$ then the sequence is rising.
For $d = 2: 1, 3, 5, 7, 9, …$
We say that the sequence is rising if the members are getting larger and larger.
If $ d < 0$ then the sequence is falling.
For $d = -2: 1, -1, -3, -5, -7, -9, …$
We can say that the sequence is falling if the members are getting smaller and smaller.
If $ d = 0$ then the sequence is a constant sequence.
For $d = 0 1, 1, 1, 1, 1, …$
General member of arithmetic sequence
To find general member of arithmetic sequence we have to observe the definition of a sequence and try to extract every member through the elements that have to be given in the task. This means that we’ll try to represent every member as a relationship between the first member and the difference.
$ a_2 = a_1 + d$
$ a_2 = a_2 + d = a_1 + 2d$
$ a_3 = a_3 + d = a_1 + 3d$
…
$ a_n = a_1 +(n – 1)d$
Using the same logic we got the nth member.
The general member of arithmetic sequence whose first member is with difference $d$ has a form: $ a_n = a_1 + (n – 1)d$
Example 6. Find tenth member of a sequence
$5, 9, 13, 17, …$
From the task itself we can see the first member $ a_1 = 5$, and calculate difference $ 9 – 5 = 13 – 9 = 17 – 13 = 4$.
$ a_1 = 5, d = 4$
Now for the tenth member we simply insert everything in the form of a general member we know.
$ a_{10} = a_1 + (10 – 1)d$
$ a_{10} =5 + 9 \cdot 4$
$ a_{10} =41$
Example 7. If fifth member of an arithmetic sequence is $14$, and the tenth is $24$, what is the first member of the sequence and what is the difference of this sequence?
$ a_5 = a_1 + 4d \Rightarrow 14 = a_1 + 4d$
$ a_{10} = a_1 + 9d \Rightarrow 24 = a_1 + 9d$
This now comes down to a simple system of equations. By the method of contrary coefficients:
$ 10 = 5d \Rightarrow d = 2$
$ 14 – 8 = a_1 \Rightarrow a_1 = 6$
Sum of first n members of arithmetic sequence:
If we have a given arithmetic sequence $ a_1, a_2, a_3,…, a_n$. If we mark the sum of first $n$ members with $S_n$ then:
$ S_1 = a_1$
$ S_2 = a_1 + a_2$
$ S_3 = a_1 + a_2 + a_3$
….
$ S_n = a_1 + a_2 + a_3 + … + a_n$
How can we determine $ S_n$ without having to add together all members one by one?
Let’s take it step by step.
$ S_n = a_1 + a_2 + a_3 + … + a_n$
$ S_n = a_n + a_{n – 1} + a_{n – 2} + … + a_1$
$ S_n = a_1 + (a_1 + d) + … +(a_1 + (n – 2)d) + (a_1 + (n – 1)d)$
$ S_n = a_1 + (a_n + d) + … +(a_n + (n – 2)d) + (a_n + (n – 1)d)$
If we add together these two equalities we get:
$ 2S_n = n (a_1 + a_n)$
$ S_n = \frac{n}{2} (a_1 + a_n)$
The sum of first $n$ members is given with a formula $ S_n = \frac{n}{2} (a_1 + a_n)$
As we know, $ a_n = a_1 + (n – 1)d$ so we can write this formula as:
$ S_n = \frac{n}{2} (2a_1 + (n – 1)d)$
Example 8. Determine the sum of the first fourteen members of the sequence $3, 6, 9, 12, 15, 18, 21, …$.
$ s_{14} = \frac{14}{2} (2\cdot 3 + 13 \cdot 3)$
$ s_{14} = 7(6 + 39)$
$ s_{14} = 7 \cdot 45$
$ s_{14} = 315$
