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Arithmetic sequence

arithmetic sequences

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

 

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