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Infinite Series

By Patricia Martin

Learn about infinite series and how to use them. Find definitions, example problems and practice problems at Thinkster Math.

Why is this concept useful?

Infinite series are used often in everyday life. Some examples of where infinite series may be used are in computer science, calculus, engineering, or physics.

Where does this concept fit into the curriculum?

Algebra I

What are Infinite series? An infinite series is  the sum of an infinite number of terms that follow a specific rule.

How to use infinite series:

An infinite series adds up all the terms of an infinite sequence and follows a set rule.

For example in the sequence 20, 40, 60….., the rule is n+20.  20 is added to each term to get the next term.

The infinite series would be 20+40+60+...

The dots at the end tell us that the sequence is infinite and will continue to keep going.

Convergent vs Divergent

An infinite series is said to be convergent if the sum of all the terms approaches a finite number.

For example:

1/3+1/9+1/27+1/81….

This sequence is adding up to get closer and closer to 1.

An infinite series is said to be divergent if the sum of all the terms  does not approach a finite number.

For example:

10,20,30,40…..

This sequence will not approach a finite number. It will approach infinity.

Geometric Series vs Arithmetic Series:

An arithmetic sequence is a sequence in which a constant is added or subtracted to each term in order to get the next term.

For example in the infinite sequence 2,5,8,11…., each term has the constant 3 added to it in order to get to the next term.

The formula to find the nth term in an arithmetic series is:

ana_n
=
a1a_1
+ (n-1)d

Where,

ana_n
= the nth term

a1a_1
= the first term of the sequence

d = the common difference between each term.

Example:

The arithmetic sequence is

81, 73, 65, 57,...

Find the value of a₃₁.

In this example,

ana_n
= the nth term

a1a_1
= 81

d = -8

n=31

Using the formula:

ana_n
=
a1a_1
+(n-1) d

We get:

ana_n
=81+(31-1)-8

ana_n
=81+-240

ana_n
=-159


Therefore, a₃₁= -159

A geometric sequence is a sequence in which each term is multiplied by a constant in order to get the next term.

For example in the infinite sequence 10,50,250,1,250…., each term is multiplied by the constant 5 in order to get to the next term.

The formula to find the nth term in a geometric series is:

ana_n
=ar⁽ⁿ⁻¹⁾

Where,

ana_n
=the nth term


a= the first term of the sequence and

r= the common ratio between each term.

Example:

Write the equation for the nth term of the geometric series  and find the value of the fourth term

-6,-42,-294,...

In this example,

ana_n
= the 4th term

a= -6

r= 7

n=4

Using the formula:

ana_n
=ar⁽ⁿ⁻¹⁾

We get:

ana_n
=-6(7)³

ana_n
=-6(343)

ana_n
=-2,058

Therefore the fourth term is -2,058.

Sample Math Problems

Question

Find the next two numbers from the arithmetic sequence 3, 3.5, 4, 4.5…..

Solution

In this problem we first need to look at what is added or subtracted to each term to find the next two terms.

Looking at the sequence, we see that the common constant added to each term is 0.5.

To find the next term, we must add 0.5 to the last term 4.5.

4.5+0.5= 5

Now to find the 6th term, we need to add 0.5 again to 5.

0.5+5=5.5

Therefore, the next two terms are 5 and 5.5.

Question

Write the equation for the nth term and find the value of a₂₅

Solution

The arithmetic sequence is 5,8,11,14….

To solve this problem use the formula:

 

ana_n
=a₁+(n-1)d

ana_n
=the nth term

a₁=5

d = 3

n=25

Plugging these numbers into the equation, we get:

ana_n
=5+(25-1)3

ana_n
=5+72

ana_n
=77

Therefore, a₂₅= 77

Question

Find the next two terms from the geometric sequence 10, 30, 90,.....

Solution

In this problem we first need to look at what is multiplied to each .

Looking at the sequence, we see that the common constant multiplied to each term is 3

To find the next term, we must multiply 90 by 3.

90 x 3= 270

Now to find the 5th term, we need to 270 by 3.

270 x 3= 810

Therefore, the next two terms are 270 and 810

Question

Write the equation for the nth term and find the value of the 5th term

The geometric sequence is 4, 20, 100,...

Solution

To solve this problem use the formula

ana_n
=ar⁽ⁿ⁻¹⁾

ana_n
= the 5th term

a= 4

r= 5

n=5

Plugging these numbers into the equation , we get:

ana_n
=4(5)⁴

ana_n
=4(625)

ana_n
=2,500

Therefore the fifth term is 2,500.

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Practice Math Problems

1.) Find the missing numbers from the arithmetic sequence.

83, 70 ,___, 44 , ____, 18…….

2.) Write the equation for the nth term of the arithmetic sequence and find the value of a₁₆,

31, 54, 77, 100, ...

3.) Find the next two terms from the geometric sequence

212, 106, 53,.....

4.) Write the equation for the nth term and find the value of the 7th term.

The geometric sequence is 36, 12, 4,...

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