Functions and Lines - Algebra

Card 0 of 8523

Question

Write a rule for the following arithmetic sequence:

$\left \{ 2,5,8,11,14,17,20\right \}$

Answer

Know that the general rule for an arithmetic sequence is

,

where represents the first number in the sequence, is the common difference between consecutive numbers, and is the -th number in the sequence.

In our problem, .

Each time we move up from one number to the next, the sequence increases by 3. Therefore, .

The rule for this sequence is therefore .

Compare your answer with the correct one above

Question

Consider the arithmetic sequence

$\left \{ 2n+4,n^{2}-3,6n,9n-7,9n+1\right \}$ .

If , find the common difference between consecutive terms.

Answer

In arithmetic sequences, the common difference is simply the value that is added to each term to produce the next term of the sequence. When solving this equation, one approach involves substituting 5 for to find the numbers that make up this sequence. For example,

so 14 is the first term of the sequence. However, a much easier approach involves only the last two terms, and .

The difference between these expressions is 8, so this must be the common difference between consecutive terms in the sequence.

Compare your answer with the correct one above

Question

Find the common difference in the following arithmetic sequence.

$\left \{ 16,32,48,64...\right \}$

Answer

An arithmetic sequence adds or subtracts a fixed amount (the common difference) to get the next term in the sequence. If you know you have an arithmetic sequence, subtract the first term from the second term to find the common difference.

Compare your answer with the correct one above

Question

Find the common difference in the following arithmetic sequence.

$\left \{ 54,27,0,-27...\right \}$

Answer

An arithmetic sequence adds or subtracts a fixed amount (the common difference) to get the next term in the sequence. If you know you have an arithmetic sequence, subtract the first term from the second term to find the common difference.

(i.e. the sequence advances by subtracting 27)

Compare your answer with the correct one above

Question

Find the next term in the following sequence.

$\left \{ ...-17, -4, 9...\right \}$

Answer

The two things we need to find out are HOW the sequence changes (adddition, subtraction, multiplication, division, etc.) and by WHAT factor.

Start by finding the difference between the first two terms.

Now let's find the difference between the 2nd and 3rd given term.

Based on these two points, we can infer that this sequence changes by adding 13 to the previous term. Therefore...

the next term in the sequence is 22.

Compare your answer with the correct one above

Question

What is the 11th term in the following sequence?

$\left \{ 87, 78, 69, 60...\right \}$

Answer

First we need to show how this sequence is changing. Let's call the first number $n_{1}$ , and the second number $n_{2}$ , and so on.

$n_{1}-n_{2}=9$

$n_{2}-n_{3}=9$

Ok, so we have established that the sequence is shrinking by 9 each time. So now we need to calculate out to the 11th term. Starting from the first term, we need to subtract 9 ten times to get to the 11th term. So that would look like this.

$n_{11}=n_{1}+(10\times -9)=87+(10\times -9)=-3$

Compare your answer with the correct one above

Question

An arithmetic sequence begins with $\left \{ -9,-4,1,6...\right \}$ . If is the first term in the sequence, find the 31st term.

Answer

For arithmetic sequences, we use the formula $a_{n} = a_{1}+(n-1)d$ , where $a_{n}$ is the term we are trying to find, $a_{1}$ is the first term, and is the difference between consecutive terms. In this case, $a_{1} = -9$ and . So, we can write the formula as $a_{31} = -9 +(30)(5)$ , and $a_{31} = 141$ .

Compare your answer with the correct one above

Question

To find any term of an arithmetic sequence:

$a_{n}=a_{1}+(n-1)d$

Where $a_{1}$ is the first term, is the number of the term to find, and is the common difference in the sequence.

Find the 18th term of the following arithmetic sequence.

$\left \{ 2,5,8,11,14...\right \}$

Answer

Start by finding the common difference, , in this sequence, which you can get by subtracting the first term from the second.

Then, using the formula given before the question:

$a_{18}=2+(18-1)3=2+(17)3=2+51=53$

Compare your answer with the correct one above

Question

To find any term of an arithmetic sequence:

$a_{n}=a_{1}+(n-1)d$

Where $a_{1}$ is the first term, is the number of the term to find, and is the common difference in the sequence.

Find the 26th term of the following arithmetic sequence.

$\left \{ 101,92,83...\right \}$

Answer

Start by finding the common difference in terms by subtracting the first term from the second.

Then, fill in the rest of the equation given before the question.

$a_{26}=101+(26-1)-9=101+(25)-9$

Compare your answer with the correct one above

Question

Each of the following 4 sets defines a relationship between and . Which of these four sets defines a one-to-one function:

A = $\left \{ \left ( 1,2 \right )\left ( 2,3 \right )\left ( 3,4 \right )\left ( 4,5 \right ) \right \}$

B= $\left \{ \left ( 1,3 \right )\left ( 2,3 \right )\left ( 3,5 \right )\left ( 4,5 \right ) \right \}$

C = $\left \{ \left ( 1,2 \right )\left ( 2,3 \right )\left ( 2,4 \right )\left ( 2,5 \right )\left ( 6,7\right )\left\right \}$

D = $\left \{ \left ( 1,7 \right )\left ( 2,7 \right )\left ( 3,7 \right )\left ( 4,7 \right ) \right \}$

Answer

Only in set A one can see that there is an unique value of for each value of and similarly each of the values maps into one and only one value. Hence set A must define a one-to-one function.

Compare your answer with the correct one above

Question

is a one-to-one function specified in terms of a set of $\left ( x,y \right )$ coordinates:

A = $\left \{ \left ( 0,0 \right ), \left ( 1,2\right ),\left ( 2,4 \right ), \left ( 3,6 \right )\right \}$

Which one of the following represents the inverse of the function specified by set A?

B = $\left \{ \left ( 0,0 \right ), \left ( 0,1 \right ), \left ( 0,2 \right ), \left ( 0,3 \right ) \right \}$

C = $\left \{ \left ( 0,0 \right ), \left ( 2,1 \right ), \left ( 4,2 \right ), \left ( 6,3 \right ) \right \}$

D = $\left \{ \left ( 1,1 \right ), \left ( 2,1 \right ), \left ( -2,1 \right ), \left ( 3,1 \right ) \right \}$

E = $\left \{ \left ( -2,1 \right ), \left ( -2,5 \right ), \left ( -2,3 \right ), \left ( -2, -2 \right ) \right \}$

F = $\left \{ \left ( 2,0 \right ), \left ( 2,-3 \right ),\left ( 2,5 \right ), \left ( 2,-7 \right ) \right \}$

Answer

The set A is an one-to-one function of the form

One can find $f^{-1}\left ( x \right )$ by interchanging the and coordinates in set A resulting in set C.

Compare your answer with the correct one above

Question

You are given a relation that comprises the following five points:

$\left \{(1,2), (3,N), (5,4), (N+3, 9), (2N-1, 10) \right \}$

For which value of is this relation a function?

Answer

A relation is a function if and only if no -coordinate is paired with more than one -coordinate. We test each of these four values of to see if this happens.

:

The points become:

Since -coordinate 1 is paired with two -coordinates, 2 and 9, the relation is not a function.

:

The points become:

Since -coordinate 3 is paired with two -coordinates, 0 and 9, the relation is not a function.

:

The points become:

Since -coordinates 3 and 5 are each paired with two different -coordinates, the relation is not a function.

:

The points become:

Since each -coordinate is paired with one and only one -coordinate, the relation is a function. is the correct choice.

Compare your answer with the correct one above

Question

Write a rule for the following arithmetic sequence:

$\left \{ 2,5,8,11,14,17,20\right \}$

Answer

Know that the general rule for an arithmetic sequence is

,

where represents the first number in the sequence, is the common difference between consecutive numbers, and is the -th number in the sequence.

In our problem, .

Each time we move up from one number to the next, the sequence increases by 3. Therefore, .

The rule for this sequence is therefore .

Compare your answer with the correct one above

Question

Consider the arithmetic sequence

$\left \{ 2n+4,n^{2}-3,6n,9n-7,9n+1\right \}$ .

If , find the common difference between consecutive terms.

Answer

In arithmetic sequences, the common difference is simply the value that is added to each term to produce the next term of the sequence. When solving this equation, one approach involves substituting 5 for to find the numbers that make up this sequence. For example,

so 14 is the first term of the sequence. However, a much easier approach involves only the last two terms, and .

The difference between these expressions is 8, so this must be the common difference between consecutive terms in the sequence.

Compare your answer with the correct one above

Question

Find the common difference in the following arithmetic sequence.

$\left \{ 16,32,48,64...\right \}$

Answer

An arithmetic sequence adds or subtracts a fixed amount (the common difference) to get the next term in the sequence. If you know you have an arithmetic sequence, subtract the first term from the second term to find the common difference.

Compare your answer with the correct one above

Question

In the following arithmetic sequence, what is ?

$\left \{ -1,n,13...\right \}$

Answer

The question states that the sequence is arithmetic, which means we find the next number in the sequence by adding (or subtracting) a constant term. We know two of the values, separated by one unknown value.

We know that is equally far from -1 and from 13; therefore is equal to half the distance between these two values. The distance between them can be found by adding the absolute values.

$\frac{(\left |13 \right |+\left | -1\right |)}{2}=arithmetic \ constant$

$\frac{14}{2}=7$

The constant in the sequence is 7. From there we can go forward or backward to find out that .

Compare your answer with the correct one above

Question

Find the next term in the following sequence.

$\left \{ ...-17, -4, 9...\right \}$

Answer

The two things we need to find out are HOW the sequence changes (adddition, subtraction, multiplication, division, etc.) and by WHAT factor.

Start by finding the difference between the first two terms.

Now let's find the difference between the 2nd and 3rd given term.

Based on these two points, we can infer that this sequence changes by adding 13 to the previous term. Therefore...

the next term in the sequence is 22.

Compare your answer with the correct one above

Question

What is the 11th term in the following sequence?

$\left \{ 87, 78, 69, 60...\right \}$

Answer

First we need to show how this sequence is changing. Let's call the first number $n_{1}$ , and the second number $n_{2}$ , and so on.

$n_{1}-n_{2}=9$

$n_{2}-n_{3}=9$

Ok, so we have established that the sequence is shrinking by 9 each time. So now we need to calculate out to the 11th term. Starting from the first term, we need to subtract 9 ten times to get to the 11th term. So that would look like this.

$n_{11}=n_{1}+(10\times -9)=87+(10\times -9)=-3$

Compare your answer with the correct one above

Question

An arithmetic sequence begins with $\left \{ -9,-4,1,6...\right \}$ . If is the first term in the sequence, find the 31st term.

Answer

For arithmetic sequences, we use the formula $a_{n} = a_{1}+(n-1)d$ , where $a_{n}$ is the term we are trying to find, $a_{1}$ is the first term, and is the difference between consecutive terms. In this case, $a_{1} = -9$ and . So, we can write the formula as $a_{31} = -9 +(30)(5)$ , and $a_{31} = 141$ .

Compare your answer with the correct one above

Question

To find any term of an arithmetic sequence:

$a_{n}=a_{1}+(n-1)d$

Where $a_{1}$ is the first term, is the number of the term to find, and is the common difference in the sequence.

Find the 18th term of the following arithmetic sequence.

$\left \{ 2,5,8,11,14...\right \}$

Answer

Start by finding the common difference, , in this sequence, which you can get by subtracting the first term from the second.

Then, using the formula given before the question:

$a_{18}=2+(18-1)3=2+(17)3=2+51=53$

Compare your answer with the correct one above

Tap the card to reveal the answer