Arithmetic sequence

MINISTRY OF EDUCATION AND SCIENCE OF THE REPUBLIC OF TAJIKISTAN

PRESIDENTIAL LYCEUM-BOARDING SCHOOL FOR THE GIFTED CHILDREN

OF THE REPUBLIC OF TAJIKISTAN

TOPIC: Arithmetic and Geometric Sequences

9 th class

Created by NUSHERVON NORMAHMEDOV

DUSHANBE-2020

What is a sequence?

• A function which is defined in the set of natural numbers is called a sequence .
• If someone asked you to list the squares of all the natural numbers, you might begin by writing 1, 4, 9, 16, 25, 36, ...
• For example, we can also express the above list of numbers by writing

f (1), f (2), f (3), f (4), f (5), f (6), ..., f ( n ), ...

where f(n) = . Here f (1) is the first term, f (2) is the second term, and so on. f(n) = is a function of n , defined in the set of natural numbers.

• However, we do not usually use functional notation to describe sequences. Instead, we denote the first term by , the second term by , and so on. So for the above list

= 1, = 4, = 9, = 16, = 25, = 36, ..., = , ...

• Here, is the first term,
• is the second term,
• is the third term,
• ...
• is the n th term, or the general term.
• Since this is just a matter of notation, we can use another letter instead of the letter a . For example, we can also use , , , etc. as the name for the general term of a sequence.

• Example 1

• Write the first five terms of the sequence with general term

=

Solution

• Since we are looking for the first five terms, we just recalculate the general term for n = 1, 2, 3, 4, 5, which gives 1, , , .

What is a arithmetic sequence?

• If a sequence () has the same difference d between its consecutive terms, then it is called an arithmetic sequence .
• In other words, () is arithmetic if = + d such than n N, d R.
• We call d the common difference of the arithmetic sequence.
• If d is positive, we say the arithmetic sequence is increasing .
• If d is negative, we say the arithmetic sequence is decreasing .

• Example 2

• State whether the following sequences are arithmetic or not. If a sequence is arithmetic, find the common difference.
• a. 7, 10, 13, 16, … b. 3, –2, –7, –12, … c. 1, 4, 9, 16, …

d. 6, 6, 6, 6, …

Solution

• a. arithmetic, d = 3 b. arithmetic, d = –5 c. not arithmetic d. arithmetic, d = 0

General Term

• If () is arithmetic, then we only know that = + d . Let us write a few terms.
• =
• = + d
• = + d = ( + d ) + d = + 2 d
• = + d = ( + 2 d ) + d = + 3 d
• = + 4 d
• ...
• = + ( n – 1) d
• This is the general term of an arithmetic sequence.

Example 3

– 3, 2, 7 are the first three terms of an arithmetic sequence (). Find the twentieth term.

Solution

We know that = –3 and d = – = – = 5. Using the general term formula,

= + ( n – 1) d

= –3 + (20 – 1) * 5 = 92.

SUM OF THE TERMS OF AN ARITHMETIC SEQUENCE

• The sum of the of first n terms of an arithmetic sequence () is

Example 4

Given an arithmetic sequence with = 2 and = 17, find .

Solution

Using the sum formula,

Example 5

Given an arithmetic sequence with = –14 and d = 5, find

Solution

Using the sum formula,

1, the geometric sequence is increasing when b 1 0 and decreasing when b 1 If 0 q b 1 b 1 0. If q   " width="640"

What is a GEOMETRIC sequence?

• If a sequence () has the same ratio q between its consecutive terms, then it is called a geometric sequence.
• In other words, () is geometric if = * q such that n N, q R.
• q is called the common ratio of the sequence.
• If q 1, the geometric sequence is increasing when b 1 0 and decreasing when b 1
• If 0 q b 1 b 1 0.
• If q

• Example 6

• State whether the following sequences are geometric or not. If a sequence is geometric, find the common ratio.
• a. 1, 2, 4, 8, … b. 3, 3, 3, 3, … c. 1, 4, 9, 16, … d. 5,-1,
• Solution

• geometric, q = 2 b. geometric, q = 1 c. not geometric

d. geometric, q=

General Term

• If () is geometric, then we only know that = * q . Let us write a few terms.

• This is the general term of a geometric sequence.

Example 7

If 100, 50, 25 are the first three terms of a geometric sequence (), find the sixth term.

Solution

We can calculate the common ratio as ,so , q =

Using the general term formula, =*, so

SUM OF THE TERMS OF A GEOMETRIC SEQUENCE

• The sum of the first n terms of a geometric sequence () is

,

Example 8

Given a geometric sequence with = and q=3, find

Solution

, so =

INFINITE SUM OF A GEOMETRIC SEQUENCE

• The infinite sum of a geometric sequence () with common ratio | q | 1 is denoted by S , and is given by the formula

S

Example 9

Find 100 + 50 + 25 + ...

Solution

Here =100 and q =. Using the infinite sum formula,