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Arithmetic Sequences Trivia Quiz

An arithmetic sequence is a sequence in which all the terms have a common difference. Knowledge of linear equations from algebra is needed. Good Luck!

A multiple-choice quiz by rodney_indy. Estimated time: 5 mins.

Author
rodney_indy

Time
5 mins

Type
Multiple Choice

Quiz #
291,144

Updated
Dec 03 21

# Qns
10

Difficulty
Average

Avg Score
6 / 10

Plays
1636

One at a Time - Single Page - Timed Game

Question 1 of 10

1. An arithmetic sequence is one in which the difference between any two consecutive terms of the sequence is constant. I will call this "common difference" d. For example, for the sequence 2, 6, 10, 14, ... we have d = 4 since 6 - 2 = 4, 10 - 6 = 4, 14 - 10 = 4, and so on. Now you try one:

What is the value of d for the following sequence:

1, 11, 21, 31, 41, ... ? Hint

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Question 2 of 10

2. The common difference -- that is, one term minus the preceding term -- in an arithmetic sequence doesn't have to be positive. For example, what is the common difference in the following arithmetic sequence?

98, 89, 80, 71, ... Hint

-9

-10

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Question 3 of 10

3. In general, we would use subscript notation to indicate the nth term of a sequence: a_n denotes the nth term of a certain sequence. For example, if I use the letter b to represent the terms in the sequence

3, 8, 13, 18, 23, ...

then b_1 = 3 (first term), b_2 = 8 (second term), and so on. So for this sequence, what is the value of b_6 ? Hint

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Question 4 of 10

4. A sequence of real numbers can be defined as a function whose domain is the set of natural numbers (the numbers 1, 2, 3, ...) and whose range is a subset of the set of real numbers. This may seem like a strange definition, but it makes perfect sense - if you input a positive integer (say 5), the output will be that term of the sequence (in this case the fifth term), and you only have one output for each input, hence a sequence of numbers really represents a function. Now let's consider these arithmetic sequences I've been describing. If we let n denote the independent variable (a positive integer), what type of function of n does an arithmetic sequence represent? Hint

an exponential function

a linear function

a quadratic function

a logarithmic function

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Question 5 of 10

5. There is a formula in general for the nth term a_n of an arithmetic sequence given the first term a_1 and the common difference d:

a_n = a_1 + (n - 1)d

For example, consider the sequence 1, 8, 15, 22, 29, ... .
Its first term a_1 = 1 and its common difference d = 7. Suppose we wanted the 100th term of this sequence - well, we'd just use the above formula:

a_100 = 1 + (100 - 1)*7 = 694.

What is the 2008th term of the above sequence? Hint

14057

14064

14071

14050

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Question 6 of 10

6. The formula a_n = a_1 + (n - 1)d can be used to give a formula for the general term of the arithmetic sequence. For example, the sequence

3, 15, 27, 39, 51, ...

has a_1 = 3 and common difference d = 12, hence a formula for the general term is given by a_n = 3 + (n - 1)*12 which simplifies:

a_n = 12n - 9

Now your turn: Which of the following is the general term b_n of the sequence

20, 18, 16, 14, 12, ... ? Hint

b_n = 20 - 2n

b_n = 18n

b_n = 18 - 2n

b_n = 22 - 2n

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Question 7 of 10

7. Now suppose I give you the formula for the general term of a certain arithmetic sequence:

a_n = 3n + 17

What is the first term a_1 of this sequence? Hint

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Question 8 of 10

8. Now let's look at other uses for the formula for the nth term of an arithmetic sequence. Suppose we are given that the first term of an arithmetic sequence is 5 and the tenth term of the sequence is 77. What is the fourteenth term of this sequence? To answer this question, we use the formula to find the common difference d:

a_n = a_1 + (n - 1)d

Substituting, you get 77 = 5 + 9d. Solving, you get d = 8. Now plug this back into the formula above to get the fourteenth term:

a_14 = 5 + (14 - 1)*8 = 109.

Now it's your turn. Suppose an arithmetic sequence has first term 1 and fifteenth term 99. What is the fourth term of this sequence? Hint

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Question 9 of 10

9. Recall that the formula for the nth term a_n of an arithmetic sequence is given by

a_n = a_1 + (n - 1)d

This time, suppose we're given that the tenth term of an arithmetic sequence is 28 and the eighteenth term of the arithmetic sequence is 44. What is the first term of this sequence? Hint

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Question 10 of 10

10. Now here's one to think about: A certain arithmetic sequence consists entirely of positive integers. For this sequence, the first term a_1 = 1 and the number 2008 appears somewhere in the sequence. How many different arithmetic sequences have these properties? You will need to make use of the formula for the nth term a_n of an arithmetic sequence: a_n = a_1 + (n - 1)d Hint

infinitely many

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Quiz Answer Key and Fun Facts

1. An arithmetic sequence is one in which the difference between any two consecutive terms of the sequence is constant. I will call this "common difference" d. For example, for the sequence 2, 6, 10, 14, ... we have d = 4 since 6 - 2 = 4, 10 - 6 = 4, 14 - 10 = 4, and so on. Now you try one: What is the value of d for the following sequence: 1, 11, 21, 31, 41, ... ?

Answer: 10

The common difference is 10 since we add 10 to any term to get the next term in the sequence.

Answer: -9

To get the next term in the sequence, subtract 9 from the previous term.

3. In general, we would use subscript notation to indicate the nth term of a sequence: a_n denotes the nth term of a certain sequence. For example, if I use the letter b to represent the terms in the sequence 3, 8, 13, 18, 23, ... then b_1 = 3 (first term), b_2 = 8 (second term), and so on. So for this sequence, what is the value of b_6 ?

Answer: 28

The common difference for the given sequence is 5. I've given the first five terms, and b_6 represents the sixth term of the sequence.

Therefore, b_6 = 23 + 5 = 28.

Answer: a linear function

Because of the common difference d, arithmetic sequences are linear functions with domain the set of natural numbers. We'll see this with an explicit example in the next question. The domain of the sequence is also referred to as the "index set". I defined a sequence as being "indexed" by the set of positive integers, but sequences can have other index sets - for example, many times we'll start counting with 0 and have a "zeroth term".

5. There is a formula in general for the nth term a_n of an arithmetic sequence given the first term a_1 and the common difference d: a_n = a_1 + (n - 1)d For example, consider the sequence 1, 8, 15, 22, 29, ... . Its first term a_1 = 1 and its common difference d = 7. Suppose we wanted the 100th term of this sequence - well, we'd just use the above formula: a_100 = 1 + (100 - 1)*7 = 694. What is the 2008th term of the above sequence?

Answer: 14050

By the formula, a_2008 = 1 + (2008 - 1)*7 = 14050.

6. The formula a_n = a_1 + (n - 1)d can be used to give a formula for the general term of the arithmetic sequence. For example, the sequence 3, 15, 27, 39, 51, ... has a_1 = 3 and common difference d = 12, hence a formula for the general term is given by a_n = 3 + (n - 1)*12 which simplifies: a_n = 12n - 9 Now your turn: Which of the following is the general term b_n of the sequence 20, 18, 16, 14, 12, ... ?

Answer: b_n = 22 - 2n

Use the formula b_n = b_1 + (n - 1)d. We know that the first term is 20 and the common difference d = -2:

b_n = 20 + (n - 1)*(-2)

Simplify the right hand side to get b_n = 22 - 2n.

7. Now suppose I give you the formula for the general term of a certain arithmetic sequence: a_n = 3n + 17 What is the first term a_1 of this sequence?

Answer: 20

Just put n = 1 into the formula for a_n to obtain the first term:

a_1 = 3*1 + 17 = 20.

8. Now let's look at other uses for the formula for the nth term of an arithmetic sequence. Suppose we are given that the first term of an arithmetic sequence is 5 and the tenth term of the sequence is 77. What is the fourteenth term of this sequence? To answer this question, we use the formula to find the common difference d: a_n = a_1 + (n - 1)d Substituting, you get 77 = 5 + 9d. Solving, you get d = 8. Now plug this back into the formula above to get the fourteenth term: a_14 = 5 + (14 - 1)*8 = 109. Now it's your turn. Suppose an arithmetic sequence has first term 1 and fifteenth term 99. What is the fourth term of this sequence?

Answer: 22

By the formula, a_15 = a_1 + (15 - 1)d. Substituting gives us the equation:

99 = 1 + 14d

Solving for d gives d = 7. Now plug this into the formula a_n = a_1 + (n - 1)d:

a_4 = 1 + (4 - 1)*7 = 1 + 21 = 22.

9. Recall that the formula for the nth term a_n of an arithmetic sequence is given by a_n = a_1 + (n - 1)d This time, suppose we're given that the tenth term of an arithmetic sequence is 28 and the eighteenth term of the arithmetic sequence is 44. What is the first term of this sequence?

Answer: 10

Here is one solution: We are told that a_10 = 28, so substituting that in the formula gives

28 = a_1 + (10 - 1)d

which simplifies to

a_1 + 9d = 28.

We are also told that a_18 = 44, so substituting that in the formula gives

44 = a_1 + (18 - 1)d

which simplifies to

a_1 + 17d = 44.

Subtracting the equation a_1 + 9d = 28 from the equatiion a_1 + 17d = 44 gives 8d = 16, thus d = 2. Subsituting d = 2 into either equation and solving for a_1 gives a_1 = 10, which is the answer.

Answer: 6

2008 is the nth term of this sequence for some n, so plug this into the formula:

2008 = 1 + (n - 1)d

Subtract 1 from both sides:

2007 = (n - 1)d

So the common difference is a divisor of the number 2007! Note that d must be positive, since I said that all terms in the sequence are positive integers. The divisors of 2007 are: 1, 3, 9, 223, 669, and 2007. So there are 6 possible values of d, hence 6 different arithmetic sequences with these properties. Here they are corresponding to each value of d:

d = 1: 1, 2, 3, 4, ... Here a_2008 = 2008

d = 3: 1, 4, 7, 10, ... Here a_670 = 2008

d = 9: 1, 10, 19, 28, ... Here a_224 = 2008

d = 223: 1, 224, 447, 670, ... Here a_10 = 2008

d = 669: 1, 670, 1339, 2008, ... Here a_4 = 2008

d = 2007: 1, 2008, 4015, 6023, ... Here a_2 = 2008

I hope you enjoyed this quiz! Thanks for playing!

Source: Author rodney_indy

This quiz was reviewed by FunTrivia editor crisw before going online.
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