Other Sequences and Series - Algebra 2
Card 1 of 84
The sum of the first n square numbers can be found using the formula 
. Find the sum of every square number between 1 and 1000.
The sum of the first n square numbers can be found using the formula
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The problem isn't asking us to add the first 1,000 square numbers, but all the square numbers from 1 to 1,000. To figure out this sum, you might need to look at a list of square numbers, or play around with large squares to find the largest one under 1,000. This ends up being 31: 
while 
, which is not between 1 and 1,000. So what we're adding is the first 31 square numbers.
This means we can plug 31 in for n in that formula:
The problem isn't asking us to add the first 1,000 square numbers, but all the square numbers from 1 to 1,000. To figure out this sum, you might need to look at a list of square numbers, or play around with large squares to find the largest one under 1,000. This ends up being 31:
This means we can plug 31 in for n in that formula:
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If the rule of some particular sequence is written as
,
find the first five terms of this sequence
If the rule of some particular sequence is written as
find the first five terms of this sequence
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The first term for the sequence is where 
. Thus,
So the first term is 4. Repeat the same thing for the second 
, third 
, fourth 
, and fifth 
terms.
We see that the first five terms in the sequence are
The first term for the sequence is where
So the first term is 4. Repeat the same thing for the second
We see that the first five terms in the sequence are
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Find the next term in the sequence:
2, 7, 17, 37, 77,...
Find the next term in the sequence:
2, 7, 17, 37, 77,...
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The sequence follows the pattern for the equation:
Therefore,
The sequence follows the pattern for the equation:
Therefore,
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Consider the following formula for a recursive sequence:
Which answer choice best represents this sequence?
Consider the following formula for a recursive sequence:
Which answer choice best represents this sequence?
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A recursive formula creates a sequence where each term is defined by the term(s) that precede it. In other words, in order to know term 12, you have to know term 11, etc.
The problem already tells us that the first term is 2. Let's find the second term.
We continue to find the rest of the terms in this way.
A recursive formula creates a sequence where each term is defined by the term(s) that precede it. In other words, in order to know term 12, you have to know term 11, etc.
The problem already tells us that the first term is 2. Let's find the second term.
We continue to find the rest of the terms in this way.
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A sequence is defined recursively as follows:
for 
Which of the following is the first positive term of the sequence?
A sequence is defined recursively as follows:
Which of the following is the first positive term of the sequence?
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Apply the rule to find the first few terms:
The first positive term of the sequence is 
.
Apply the rule to find the first few terms:
The first positive term of the sequence is
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Which of the following expressions describes the sequence below:
Which of the following expressions describes the sequence below:
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In order to determine which expression describes the sequence in the problem, we must determine the relationship each entry has with its position in the sequence. For example, for n=1, we must determine which expression involving n will yield a result of 3, for n=2, we must determine which expression will yield a result of 8, and so on, ensuring that the expression holds true for every n value in the sequence. If we check each of our answers, we can see that only the following expression will give the correct result for each increasing value of n:
If we continue, we can see that we will obtain the sequence 3,8,15,24,35,48,63,..., so this expression is the correct representation of the sequence given in the problem.
In order to determine which expression describes the sequence in the problem, we must determine the relationship each entry has with its position in the sequence. For example, for n=1, we must determine which expression involving n will yield a result of 3, for n=2, we must determine which expression will yield a result of 8, and so on, ensuring that the expression holds true for every n value in the sequence. If we check each of our answers, we can see that only the following expression will give the correct result for each increasing value of n:
If we continue, we can see that we will obtain the sequence 3,8,15,24,35,48,63,..., so this expression is the correct representation of the sequence given in the problem.
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What is the mean of the following quiz scores.
What is the mean of the following quiz scores.
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To find the mean of a set of numbers we first must add all the numbers together.
Using the formula for mean we get,
Therefore we get,
To find the mean of a set of numbers we first must add all the numbers together.
Using the formula for mean we get,
Therefore we get,
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What is the median of the following prices.
What is the median of the following prices.
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The median of a set of numbers is the middle value of the set.
To find the middle value of this particular data set put the prices in order of lowest to highest
Since we have an even number of entries we will need to find the mean of the two middle numbers and this will become our median.
The median of a set of numbers is the middle value of the set.
To find the middle value of this particular data set put the prices in order of lowest to highest
Since we have an even number of entries we will need to find the mean of the two middle numbers and this will become our median.
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Complete the following sequences.
Complete the following sequences.
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The sequence goes down by 2 so,
.
Therefore the next number in the sequence is 
.
The sequence goes down by 2 so,
Therefore the next number in the sequence is
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Complete the following sequence
Complete the following sequence
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The sequence goes up by 5 so,
Therefore the next term in the sequence will be 
.
The sequence goes up by 5 so,
Therefore the next term in the sequence will be
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What is 
percent of 
?
What is
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To find the value related to the specific percentage we need to set up a proportion and solve for x.
From here we cross multiply and divide to find the value of x.
To find the value related to the specific percentage we need to set up a proportion and solve for x.
From here we cross multiply and divide to find the value of x.
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What is 
percent of 
?
What is
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To find the value for a specific percentage of a number we first need to convert the percentage into a decimal.
From here we multiply the decimal with the number we are given in the question.
To find the value for a specific percentage of a number we first need to convert the percentage into a decimal.
From here we multiply the decimal with the number we are given in the question.
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The harmonic series is 
where the nth term is the reciprocal of n. Which would work as a recursive formula 
where 
is the nth term?
The harmonic series is
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To go from 
to 
, we're adding 1 to the denominator. In words, we're flipping 
, adding 1, then flipping it again. For example, to get from 
to 
we would have to flip 
to be 4, add 1 to get 5, then flip again to get 
.
The formula that shows this is
To go from
The formula that shows this is
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The sum of the first n integerss can be found using the formula 
.
Find the sum of every number between 17 and 8,043, inclusive.
The sum of the first n integerss can be found using the formula
Find the sum of every number between 17 and 8,043, inclusive.
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To find the sum of all the integers in between 17 and 8,043, first we will find the sum of every integer from 8,043, and then we will subtract out the sum of the numbers 1-16, since those aren't between 17 and 8,043.
The sum of the first 8,043 integers is 
The sum of the integers 1-16 is 
Subtracting gives us 
To find the sum of all the integers in between 17 and 8,043, first we will find the sum of every integer from 8,043, and then we will subtract out the sum of the numbers 1-16, since those aren't between 17 and 8,043.
The sum of the first 8,043 integers is
The sum of the integers 1-16 is
Subtracting gives us
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The sum of the first n integers can be found using the formula 
Find the sum of all the integers from -2,256 to 4,400.
The sum of the first n integers can be found using the formula
Find the sum of all the integers from -2,256 to 4,400.
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To calculate this sum, first we will need to find the sum of the positive integers, then the negative interers, then add them together.
To find the sum of the positive integers, use the formula with 
:
To find the sum of the negative integers, we can use the same formula as the positive numbers and then just make that answer negative.
so the negative numbers add up to 
.
The final answer is 
To calculate this sum, first we will need to find the sum of the positive integers, then the negative interers, then add them together.
To find the sum of the positive integers, use the formula with
To find the sum of the negative integers, we can use the same formula as the positive numbers and then just make that answer negative.
The final answer is
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Evaluate:
Evaluate:
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is equal to the sum of the expressions formed by substituting 1, 2, 3, 4, and 5, in turn, for 
in the expression 
. This is simply the sum of the reciprocals of these 5 integers, which is equal to
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A sequence begins as follows:
Which statement is true?
A sequence begins as follows:
Which statement is true?
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An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the differences between each term and the previous term is not constant from term to term:
The sequence cannot be arithmetic.
A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratios of each term to the previous one is not constant from term to term:
The sequence cannot be geometric.
An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the differences between each term and the previous term is not constant from term to term:
The sequence cannot be arithmetic.
A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratios of each term to the previous one is not constant from term to term:
The sequence cannot be geometric.
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A sequence begins as follows:
Which statement is true?
A sequence begins as follows:
Which statement is true?
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An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term varies from term to term:
The sequence cannot be arithmetic.
A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one also varies from term to term:
The sequence cannot be geometric.
An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term varies from term to term:
The sequence cannot be arithmetic.
A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one also varies from term to term:
The sequence cannot be geometric.
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A sequence begins as follows:
Which statement is true?
A sequence begins as follows:
Which statement is true?
Tap to reveal answer
An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term varies from term to term:
The sequence cannot be arithmetic.
A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one varies from term to term:
The sequence cannot be geometric.
An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term varies from term to term:
The sequence cannot be arithmetic.
A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one varies from term to term:
The sequence cannot be geometric.
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A sequence begins as follows:
Which statement is true?
A sequence begins as follows:
Which statement is true?
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An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term varies from term to term:
The first difference: 
The second difference: 
The sequence cannot be arithmetic.
A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one varies from term to term:
The first ratio: 
The second ratio: 
The sequence cannot be geometric.
An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term varies from term to term:
The first difference:
The second difference:
The sequence cannot be arithmetic.
A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one varies from term to term:
The first ratio:
The second ratio:
The sequence cannot be geometric.
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