Sequences and Series - Math
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Indicate the first three terms of the following series:
Indicate the first three terms of the following series:
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In the arithmetic series, the first terms can be found by plugging 
, 
, and 
into the equation.
In the arithmetic series, the first terms can be found by plugging
Indicate the first three terms of the following series:
Indicate the first three terms of the following series:
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In the arithmetic series, the first terms can be found by plugging in 
, 
, and 
for 
.
In the arithmetic series, the first terms can be found by plugging in
Find the sum of all even integers from 
to 
.
Find the sum of all even integers from
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The formula for the sum of an arithmetic series is
,
where 
is the number of terms in the series, 
is the first term, and 
is the last term.
The formula for the sum of an arithmetic series is
where
Find the sum of all even integers from 
to 
.
Find the sum of all even integers from
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The formula for the sum of an arithmetic series is
,
where 
is the number of terms in the series, 
is the first term, and 
is the last term.
We know that there are 
terms in the series. The first term is 
and the last term is 
. Our formula becomes:
The formula for the sum of an arithmetic series is
where
We know that there are
Find the sum of the even integers from 
to 
.
Find the sum of the even integers from
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The sum of even integers represents an arithmetic series.
The formula for the partial sum of an arithmetic series is
,
where 
is the first value in the series, 
is the number of terms, and 
is the difference between sequential terms.
Plugging in our values, we get:
The sum of even integers represents an arithmetic series.
The formula for the partial sum of an arithmetic series is
where
Plugging in our values, we get:
Find the value for 
Find the value for
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To best understand, let's write out the series. So
We can see this is an infinite geometric series with each successive term being multiplied by 
.
A definition you may wish to remember is
where 
stands for the common ratio between the numbers, which in this case is 
or 
. So we get
To best understand, let's write out the series. So
We can see this is an infinite geometric series with each successive term being multiplied by
A definition you may wish to remember is
Evaluate:
Evaluate:
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This is a geometric series whose first term is 
and whose common ratio is 
. The sum of this series is:
This is a geometric series whose first term is
Evaluate:
Evaluate:
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This is a geometric series whose first term is 
and whose common ratio is 
. The sum of this series is:
This is a geometric series whose first term is
Consider the sequence: 
What is the fifteenth term in the sequence?
Consider the sequence:
What is the fifteenth term in the sequence?
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The sequence can be described by the equation 
, where 
is the term in the sequence.
For the 15th term, 
.
The sequence can be described by the equation
For the 15th term,
What is the sixth term when 
is expanded?
What is the sixth term when
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We will need to use the Binomial Theorem in order to solve this problem. Consider the expansion of 
, where n is an integer. The rth term of this expansion is given by the following formula:
,
where 
is a combination. In general, if x and y are nonnegative integers such that x > y, then the combination of x and y is defined as follows: 
.
We are asked to find the sixth term of 
, which means that in this case r = 6 and n = 10. Also, we will let 
and 
. We can now apply the Binomial Theorem to determine the sixth term, which is as follows:
Next, let's find the value of 
. According to the definition of a combination,
.
Remember that, if n is a positive integer, then 
. This is called a factorial.
Let's go back to simplifying 
.
The answer is 
.
We will need to use the Binomial Theorem in order to solve this problem. Consider the expansion of
where
We are asked to find the sixth term of
Next, let's find the value of
Remember that, if n is a positive integer, then
Let's go back to simplifying
The answer is
What are the first three terms in the series?
What are the first three terms in the series?
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To find the first three terms, replace 
with 
, 
, and 
.
The first three terms are 
, 
, and 
.
To find the first three terms, replace
The first three terms are
Find the first three terms in the series.
Find the first three terms in the series.
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To find the first three terms, replace with 
, 
, and 
.
The first three terms are 
, 
, and 
.
To find the first three terms, replace
The first three terms are
Indicate the first three terms of the following series:
Indicate the first three terms of the following series:
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The first terms can be found by substituting 
, 
, and 
for 
into the sum formula.
The first terms can be found by substituting
Indicate the first three terms of the following series.
Indicate the first three terms of the following series.
Tap to see back →
The first terms can be found by substituting 
, 
, and 
in for 
.
The first terms can be found by substituting
Evaluate: 
Evaluate:
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This sum can be determined using the formula for the sum of an infinite geometric series, with initial term 
and common ratio 
:
This sum can be determined using the formula for the sum of an infinite geometric series, with initial term
The fourth term in an arithmetic sequence is -20, and the eighth term is -10. What is the hundredth term in the sequence?
The fourth term in an arithmetic sequence is -20, and the eighth term is -10. What is the hundredth term in the sequence?
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An arithmetic sequence is one in which there is a common difference between consecutive terms. For example, the sequence {2, 5, 8, 11} is an arithmetic sequence, because each term can be found by adding three to the term before it.
Let 
denote the nth term of the sequence. Then the following formula can be used for arithmetic sequences in general:
, where d is the common difference between two consecutive terms.
We are given the 4th and 8th terms in the sequence, so we can write the following equations:
.
We now have a system of two equations with two unknowns:
Let us solve this system by subtracting the equation 
from the equation 
. The result of this subtraction is
.
This means that d = 2.5.
Using the equation 
, we can find the first term of the sequence.
Ultimately, we are asked to find the hundredth term of the sequence.
The answer is 220.
An arithmetic sequence is one in which there is a common difference between consecutive terms. For example, the sequence {2, 5, 8, 11} is an arithmetic sequence, because each term can be found by adding three to the term before it.
Let
We are given the 4th and 8th terms in the sequence, so we can write the following equations:
We now have a system of two equations with two unknowns:
Let us solve this system by subtracting the equation
This means that d = 2.5.
Using the equation
Ultimately, we are asked to find the hundredth term of the sequence.
The answer is 220.
Find the sum, if possible:
Find the sum, if possible:
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The formula for the summation of an infinite geometric series is
,
where 
is the first term in the series and 
is the rate of change between succesive terms. The key here is finding the rate, or pattern, between the terms. Because this is a geometric sequence, the rate is the constant by which each new term is multiplied.
Plugging in our values, we get:
The formula for the summation of an infinite geometric series is
where
Plugging in our values, we get:
Find the sum, if possible:
Find the sum, if possible:
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The formula for the summation of an infinite geometric series is
,
where 
is the first term in the series and 
is the rate of change between succesive terms in a series
Because the terms switch sign, we know that the rate must be negative.
Plugging in our values, we get:
The formula for the summation of an infinite geometric series is
where
Because the terms switch sign, we know that the rate must be negative.
Plugging in our values, we get:
Find the sum, if possible:
Find the sum, if possible:
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The formula for the summation of an infinite geometric series is
,
where 
is the first term in the series and 
is the rate of change between succesive terms in a series.
In order for an infinite geometric series to have a sum, 
needs to be greater than 
and less than 
, i.e. 
.
Since 
, there is no solution.
The formula for the summation of an infinite geometric series is
where
In order for an infinite geometric series to have a sum,
Since
Determine the summation notation for the following series:
Determine the summation notation for the following series:
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The series is a geometric series. The summation notation of a geometric series is
,
where 
is the number of terms in the series, 
is the first term of the series, and 
is the common ratio between terms.
In this series, is 
, is 
, and is 
. Therefore, the summation notation of this geometric series is:
This simplifies to:
The series is a geometric series. The summation notation of a geometric series is
where
In this series,
This simplifies to: