Other Sequences and Series - Algebra II

Card 0 of 84

Question

If the rule of some particular sequence is written as

,

find the first five terms of this sequence

Answer

The first term for the sequence is where . Thus,

So the first term is 4. Repeat the same thing for the second $\left ( n=2 \right )$ , third $\left ( n=3 \right )$ , fourth $\left ( n=4 \right )$ , and fifth $\left ( n=5 \right )$ terms.

We see that the first five terms in the sequence are

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Question

Find the next term in the sequence:

2, 7, 17, 37, 77,...

Answer

The sequence follows the pattern for the equation:

$n_x=2n_{x-1}+3$

$n_{x-1}=77$

Therefore,

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Question

Consider the following formula for a recursive sequence:

$a_n=(a_{n-1})^2; a_1=2$

Which answer choice best represents this sequence?

Answer

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.

$a_n=(a_{n-1})^2; a_1=2$

The problem already tells us that the first term is 2. Let's find the second term.

$a_2=(a_{2-1})^2=(a_1)^2=2^2=4$

We continue to find the rest of the terms in this way.

$a_3=(a_{3-1})^2=(a_2)^2=4^2=16$

$a_4=(a_{4-1})^2=(a_3)^2=16^2=256$

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Question

A sequence is defined recursively as follows:

$a_{1} = 1$

$a_{2} = -1$

$a_{n} = a_{n - 1} a_{n - 2}$ for

How many of the first twenty terms of the sequence are positive?

Answer

Apply the rule to find the first few terms:

$a_{1} = 1$

$a_{2} = -1$

$a_{3} = a_{2} a_{1} = -1 \cdot 1 = -1$

$a_{4} = a_{3} a_{2} = -1 \cdot \left (-1 \right ) = 1$

$a_{5} = a_{4} a_{3} = 1 \cdot \left (-1 \right ) = - 1$

$a_{6} = a_{5} a_{4} = -1 \cdot 1 = - 1$

After the sixth term, it is apparent that this cycle will repeat itself, so the first twenty terms of the sequence will be, in order:

$\left { 1, -1, -1, 1, -1, -1,1, -1, -1, 1, -1, -1, 1, -1, -1,1, -1, -1,1, -1\right }$

Seven of these first twenty terms are positive.

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Question

A sequence is defined recursively as follows:

$a_{1} = -5$

$a_{n} = 2 a_{n-1} +6$ for

Which of the following is the first positive term of the sequence?

Answer

Apply the rule to find the first few terms:

$a_{1} = -5$

$a_{2} = 2 a_{ 1} +6 = 2 (-5) + 6 = -10 + 6 = -4$

$a_{3} = 2 a_{ 2} +6 = 2 (-4) + 6 = -8 + 6 = -2$

$a_{4} = 2 a_{ 3} +6 = 2 (-2) + 6 = -4 + 6 = 2$

The first positive term of the sequence is $a_{4}$ .

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Question

Which of the following expressions describes the sequence below:

Answer

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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Question

What is the mean of the following quiz scores.

Answer

To find the mean of a set of numbers we first must add all the numbers together.

Using the formula for mean we get,

$\text{Mean}=\frac{\text{Sum of Numbers}}{\text{Total Number of Entries}}$

Therefore we get,

$\text{Mean}=\frac{446}{6}=77.66=77.7$

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Question

What is the median of the following prices.

Answer

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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Question

Complete the following sequences.

$7, 5, 3, 1, \cdots$

Answer

The sequence goes down by 2 so,

.

Therefore the next number in the sequence is .

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Question

Complete the following sequence

$15, 20, 25, 30, 35, \cdots$

Answer

The sequence goes up by 5 so,

Therefore the next term in the sequence will be .

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Question

What is percent of ?

Answer

To find the value related to the specific percentage we need to set up a proportion and solve for x.

$\frac{15}{100}=\frac{x}{60}$

From here we cross multiply and divide to find the value of x.

$15 \times 60 = 900$

$\frac{900}{100}=9$

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Question

What is percent of ?

Answer

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.

$0.5 \times 135 = 67.5 = 68$

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Question

The harmonic series is $1, \frac{1}{2} , \frac{1}{3} , \frac{1}{4} , ... , \frac{1}{n}$ where the nth term is the reciprocal of n. Which would work as a recursive formula $a _{n} = f(a_{n-1})$ where $a_{n}$ is the nth term?

Answer

To go from $a_{n-1}$ to $a_{n}$ , we're adding 1 to the denominator. In words, we're flipping $a_{n-1}$ , adding 1, then flipping it again. For example, to get from $a_{4}=\frac{1}{4}$ to $a_{5 } = \frac{1}{5}$ we would have to flip $\frac{1}{4}$ to be 4, add 1 to get 5, then flip again to get $\frac{1}{5}$ .

The formula that shows this is

$a_{n} =\frac{1}{\frac{1}{a_{n-1}}+1}$

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Question

The sum of the first n square numbers can be found using the formula $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$ . Find the sum of every square number between 1 and 1000.

Answer

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:

$\frac{31(31+1)(312 + 1 )}{6} = \frac{3132*63}{6 } = 10,416$

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Question

The sum of the first n integerss can be found using the formula $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$ .

Find the sum of every number between 17 and 8,043, inclusive.

Answer

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 $\frac{(8043)(8044)}{2 } = 32, 348, 946$

The sum of the integers 1-16 is $\frac{(16)(17)}{2 } = 136$

Subtracting gives us

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Question

The sum of the first n integers can be found using the formula $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$

Find the sum of all the integers from -2,256 to 4,400.

Answer

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 :

$\frac{(4400)(4401)}{2 } = 9,682, 200$

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.

$\frac{(2256)(2257)}{2 } = 2,545, 896$ so the negative numbers add up to .

The final answer is

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Question

Evaluate:

$\sum_{k = 1} ^{5} \frac{1}{k}$

Answer

$\sum_{k = 1} ^{5} \frac{1}{k}$ is equal to the sum of the expressions formed by substituting 1, 2, 3, 4, and 5, in turn, for in the expression $\frac{1}{k}$ . This is simply the sum of the reciprocals of these 5 integers, which is equal to

$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}+ \frac{1}{5} = 2\frac{17}{60}$

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Question

A sequence begins as follows:

Which statement is true?

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 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:

$\frac{-16}{8} = -2$

$\frac{24}{-16} = -\frac{3}{2}$

The sequence cannot be geometric.

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Question

A sequence begins as follows:

$\frac{1}{5} , \frac{1}{6} , \frac{1}{7} , \frac{1}{8} , ...$

Which statement is true?

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:

$\frac{1}{6} - \frac{1}{5} = \frac{5}{30} - \frac{6}{30} = -\frac{1}{30}$

$\frac{1}{7} - \frac{1}{6} = \frac{6}{42} - \frac{7}{42} = -\frac{1}{42}$

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:

$\frac{1}{6} \div \frac{1}{5} =\frac{1}{6} \times \frac{5} {1}= \frac{5}{6}$

$\frac{1}{7} \div \frac{1}{6} =\frac{1}{7} \times \frac{6} {1}= \frac{6}{7}$

The sequence cannot be geometric.

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Question

A sequence begins as follows:

Which statement is true?

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:

$\frac{1}{1} = 1$

$\frac{2}{1}= 2$

The sequence cannot be geometric.

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