Algebra II : Other Sequences and Series

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1 : Other Sequences And Series

Consider the following formula for a recursive sequence:

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

Which answer choice best represents this sequence?

Possible Answers:

2, 4, 16, 32, ...

2, 4, 6, 8, ...

2, 4, 8, 16, ....

2, 4, 16, 256, ...

Correct answer:

2, 4, 16, 256, ...

Explanation:

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.

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

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

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

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

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

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

Example Question #2 : Other Sequences And Series

A sequence is defined recursively as follows:

\(\displaystyle a_{1} = 1\)

\(\displaystyle a_{2} = -1\)

\(\displaystyle a_{n} = a_{n - 1} a_{n - 2}\) for \(\displaystyle n = 3, 4, 5, ...\)

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

Possible Answers:

\(\displaystyle 7\)

\(\displaystyle 13\)

\(\displaystyle 5\)

\(\displaystyle 10\)

\(\displaystyle 1\)

Correct answer:

\(\displaystyle 7\)

Explanation:

Apply the rule to find the first few terms:

\(\displaystyle a_{1} = 1\)

\(\displaystyle a_{2} = -1\)

\(\displaystyle a_{3} = a_{2} a_{1} = -1 \cdot 1 = -1\)

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

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

\(\displaystyle a_{6} = a_{5} a_{4} = -1 \cdot 1 = - 1\)

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:

\(\displaystyle \left \{ 1, -1, -1, 1, -1, -1,1, -1, -1, 1, -1, -1, 1, -1, -1,1, -1, -1,1, -1\right \}\)

Seven of these first twenty terms are positive.

Example Question #3 : Other Sequences And Series

A sequence is defined recursively as follows:

\(\displaystyle a_{1} = -5\)

\(\displaystyle a_{n} = 2 a_{n-1} +6\) for \(\displaystyle n = 2, 3, 4,...\)

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

Possible Answers:

\(\displaystyle a_{5}\)

\(\displaystyle a_{3}\)

The sequence has no positive terms.

\(\displaystyle a_{4}\)

\(\displaystyle a_{6}\)

Correct answer:

\(\displaystyle a_{4}\)

Explanation:

Apply the rule to find the first few terms:

\(\displaystyle a_{1} = -5\)

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

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

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

The first positive term of the sequence is \(\displaystyle a_{4}\).

 

Example Question #1 : Other Sequences And Series

Which of the following expressions describes the sequence below:

\(\displaystyle 3,8,15,24,35,48,63,...\)

Possible Answers:

\(\displaystyle 3n-1\)

\(\displaystyle 2n+2\)

\(\displaystyle 5n^2-3n+1\)

\(\displaystyle n^3+3\)

\(\displaystyle n^2+2n\)

Correct answer:

\(\displaystyle n^2+2n\)

Explanation:

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:

\(\displaystyle n^2+2n\)

\(\displaystyle (1)^2+2(1)=3\)

\(\displaystyle (2)^2+2(2)=8\)

\(\displaystyle (3)^2+2(3)=15\)

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.

Example Question #3 : Other Sequences And Series

What is the mean of the following quiz scores.

\(\displaystyle 99, 70, 84, 63, 91, 59\)

Possible Answers:

\(\displaystyle 80\)

\(\displaystyle 77.7\)

\(\displaystyle 78.6\)

\(\displaystyle 76.7\)

\(\displaystyle 75\)

Correct answer:

\(\displaystyle 77.7\)

Explanation:

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

\(\displaystyle 99 + 70 + 84 + 63 + 91 + 59 = 466\)

Using the formula for mean we get,

\(\displaystyle \text{Mean}=\frac{\text{Sum of Numbers}}{\text{Total Number of Entries}}\)

 

Therefore we get,

\(\displaystyle \text{Mean}=\frac{446}{6}=77.66=77.7\)

Example Question #1 : Other Sequences And Series

What is the median of the following prices.

\(\displaystyle 83, 52, 36, 57\)

 

Possible Answers:

\(\displaystyle 54.5\)

\(\displaystyle 53.5\)

\(\displaystyle 50\)

\(\displaystyle 55.5\)

\(\displaystyle 56\)

Correct answer:

\(\displaystyle 54.5\)

Explanation:

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

\(\displaystyle 36, 52, 57, 83\)

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.

\(\displaystyle 52 + 57 = 109\)

\(\displaystyle 109 / 2 = 54.5\)

Example Question #1 : Other Sequences And Series

Complete the following sequences.

\(\displaystyle 7, 5, 3, 1, \cdots\)

Possible Answers:

\(\displaystyle 1\)

\(\displaystyle -2\)

\(\displaystyle -1\)

\(\displaystyle -3\)

\(\displaystyle 0\)

Correct answer:

\(\displaystyle -1\)

Explanation:

The sequence goes down by 2 so,

 \(\displaystyle 7 - 2 = 5 - 2 = 3 - 2 = 1 - 2 = -1\).

Therefore the next number in the sequence is \(\displaystyle -1\).

Example Question #2 : Other Sequences And Series

Complete the following sequence

\(\displaystyle 15, 20, 25, 30, 35, \cdots\)

Possible Answers:

\(\displaystyle 45\)

\(\displaystyle 5\)

\(\displaystyle 39\)

\(\displaystyle 40\)

\(\displaystyle 42\)

Correct answer:

\(\displaystyle 40\)

Explanation:

The sequence goes up by 5 so,

 \(\displaystyle 15 + 5 = 20 + 5 = 25 + 5 = 30 + 5 = 35 + 5 = 40\)

Therefore the next term in the sequence will be \(\displaystyle 40\).

Example Question #3 : Other Sequences And Series

What is \(\displaystyle 15\) percent of \(\displaystyle 60\)?

Possible Answers:

\(\displaystyle 10\)

\(\displaystyle 100\)

\(\displaystyle 900\)

\(\displaystyle 9\)

\(\displaystyle 90\)

Correct answer:

\(\displaystyle 9\)

Explanation:

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

\(\displaystyle \frac{15}{100}=\frac{x}{60}\)

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

\(\displaystyle 15 \times 60 = 900\)

\(\displaystyle \frac{900}{100}=9\)

Example Question #10 : Other Sequences And Series

What is \(\displaystyle 50\) percent of \(\displaystyle 135\)?

Possible Answers:

\(\displaystyle 66\)

\(\displaystyle 68\)

\(\displaystyle 69\)

\(\displaystyle 67\)

\(\displaystyle 70\)

Correct answer:

\(\displaystyle 68\)

Explanation:

To find the value for a specific percentage of a number we first need to convert the percentage into a decimal.

\(\displaystyle 50/100 = 0.5\)

From here we multiply the decimal with the number we are given in the question.

\(\displaystyle 0.5 \times 135 = 67.5 = 68\)

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