Mathematics High School

## Answers

**Answer 1**

Answer: The 10th term is 19

Step-by-step explanation:

For a sequence to be an arithmetic sequence, then there exist a common difference.

How do we calculate common difference?

Common difference is calculated by subtracting the first term from the second term, subtracting the second term from the third term and so on.

From the sequence given , 3 + x, 9 + 3x, 13 + 4x, .. , calculating common difference implies

(9+3x) – ( 3 +x) = 13 + 4x – ( 9 + 3x)

Expanding, we have

9 + 3x – 3 – x = 13 + 4x – 9 – 3x

⇒9 – 3+3x –x = 13 – 9 +4x – 3x

⇒6 + 2x = 4 + x

Collecting the like terms , we have

2x – x = 4 – 6

which gives X = -2

Since the value of x is gotten now , we can get what the given sequence looks like by substituting the value of x into each term given, the sequence becomes:

3 + (-2), 9 + 3(-2), 13 + 4(-2) +...

Which gives

1 , 3 ,5, …

We can therefore note that the common difference is 2

check: 3 - 1 = 5 -3 = 2

(b) To calculate the tenth term , we use the general formula for calculating nth term , which is given as

= a + (n-1)d

Where n is the number of terms , a is the first term and d is the common difference,

Therefore, = 1 + ( 10-1)x2

= 1 + 18

= 19

Therefore the 10th term is 19

## Related Questions

Find an explicit form f(n) of the arithmetic sequence where the 2nd term is 25 and the sum of the 3rd term and 4th term is 86.

### Answers

Answer:f(n) = 12n + 1

Step-by-step explanation:

Since it is an arithmetic sequence, the general formula for nth term is given as

f(n) = a + ( n-1)d , where a is the first term, n is the number of terms and d is the common difference.

Given from the question

Second term is 25 , which means that

a + d = 25

Also given , the sum of third and fifth term is 86, which means

a + 2d + a + 3d = 86

2a + 5d = 86

Combining the two equations , we have

a + d = 25 ………….. I

2a + 5d = 86………..II

Using substitution method to solve the resulting simultaneous equation

From equation I make a the subject of the formula, which gives

a = 25 – d…………………. III

Substitute the value of a into equation II , we have

2 ( 25 – d) + 5d = 86

Expanding

50 + 2d + 5d = 86

50 + 3d = 86

Collect the like terms

3d = 86 – 50

3d = 36

d = 12

substitute the value of d into equation III, we have

a = 25 – 12

a = 13

Since we have gotten the value of a and b , we will substitute into the general formula for the nth term

f(n) = a + ( n-1)

f(n) = 13 + (n-1)12

Expanding

f(n) = 13+ 12n -12

f(n) =12n + 1

Therefore the explicit form f(n) of the arithmetic sequence is f(n) =12n + 1

Find an explicit form f(n) for each of the following arithmetic sequences (assume a is some real number and x is some real number).

d. a, 2a + 1, 3a + 2, 4a + 3, ...

### Answers

Answer:

The explicit form f(n) is

Step-by-step explanation:

A sequence is arithmetic if the difference between each consecutive terms is a constant.

The explicit formula of an arithmetic sequence is:

Where d is the common difference, a1 is the first term of the sequence and an is the nth term.

In order to obtain the common difference, you have to subtract two consecutive terms:

For the first and second terms:

2a+1 - a = a+1

For the second and third terms:

3a+2 - (2a+1)=

Applying the distributive property:

3a+2-2a-1= a+1

Notice that the difference of any two consecutive terms is the same. Therefore:

d=a+1

Also, a1=a (The first term)

Replacing in the explicit formula:

Find an explicit form f(n) for each of the following arithmetic sequences (assume a is some real number and x is some real number).

b. 1 / 5, 1 / 10, 0, − 1 / 10, ...

### Answers

Answer:

The explicit form for this sequence is for all .

Step-by-step explanation:

The explicit form of an arithmetic sequence of numbers is given by the formula , where is the first term of the sequence, is the difference between two consecutive terms of the sequence, and .

We know that the first four elements for the arithmetic sequence are .

To find the general formula for this problem we only need to calculate in the above formula.

For n=2, we have

if we replace and and solve for we obtain

Therefore the explicit form is for all .

Consider the arithmetic sequence 13, 24, 35, .... a. Find an explicit form for the sequence in terms of n.

### Answers

Answer:

The explicit form for the sequence is:

Step-by-step explanation:

In order to find an explicit form for the given sequence, you have to use the definition of arithmetic sequence and the explicit formula.

An arithmetic sequence is defined as a sequence where the difference of two consecutive terms is a constant.

The explicit formula is:

Where a1 is the first term, d is the common difference and an is the nth term of the sequence.

You have to subtract two consecutive terms to obtain d:

24-13= 11

35-24=11

Therefore d=11

In this case a1=13

Replacing in the formula:

Find an explicit form f(n) for each of the following arithmetic sequences (assume a is some real number and x is some real number).

c. x + 4, x + 8, x + 12, x + 16, ...

### Answers

Answer:

Explicit form, f(n) = x +4n

Step-by-step explanation:

Here the sequence is given as x + 4, x + 8, x + 12, x + 16, ...

First term, f = x + 4

Common difference, d = x + 8 - (x+4) = 4

We have equation for nth term of arithmetic sequence as

Explicit form, f(n) = x +4n

The ratio of the current ages of two RSM summer camp students Dylan and Mike is 3:4. If Dylan had been 6 years younger and Mike 6 years older, the age of Mike would have been 6 times the age of Dylan. Arrange the ratios of Dylan's current age to Mike's current age, their ages in four years from now and those 4 years ago, in descending order.

### Answers

Answer:

13:16, 3:4, 5:8.

Step-by-step explanation:

The ratio of the current ages of two students Dylan and Mike is 3:4.

Let their ages are 3x and 4x respectively.

If Dylan had been 6 years younger and Mike 6 years older, the age of Mike would have been 6 times the age of Dylan.

Hence, we can write, 6(3x - 6) = (4x + 6)

⇒ 18x - 4x = 36 + 6

⇒14x = 42

⇒ x = 3

Therefore, the current age of Dylan is 3×3 =9 years and that of Mike is 3×4 =12 years.

Therefore, the ratio of Dylan's current age to Mike's current age is 3:4. The ratio of their ages in four years ago will be (9-4):(12-4) =5:8

The ratio of their ages in four years from now will be (9+4):(12+4) =13:16

Hence, the ratios in descending order is 13:16, 3:4, 5:8.

(Answer)

A merchant doing business in Lucca doubled his money there and then spent 12 denarii. On leaving, he went to Florence, where he also doubled his money and spent 12 denarii. Returning home to Pisa, he there doubled his money and again spent 12 denarii, nothing remaining. How much did he have in the beginning?

### Answers

Answer:

10.5 denarii

Step-by-step explanation:

Step 1: Initially Doubled his money in Lucca and spent 12 denarii

Let's assume the initial money he had before he did anything with it is x,

meaning if he doubled it in Lucca he would have (2×x)=2x, he then spent 12 denarii meaning he was left with (2x-12) denarii.

Step 2: In Florence he doubled what was left on leaving Lucca and spent 12

This is expressed as 2(2x-12)-12=4x-24-12=4x-36

(4x-36) denarii is what remained after he left Florence to Pisa

Step 3: In Pisa he doubled what was left on leaving Florence and spent 12 denarii

This is expressed as 2(4x-36)-12=8x-72-12=8x-84

Step 4: After getting back home in Pisa he was left with nothing

This is expressed as; 8x-84=0, 8x=84

solving for x by dividing both sides of the equation by 8

8x/8=84/8

x=10.5 denarii

Suppose n + 1 numbers are selected from {1, 2, 3, . . . , 2n}. Using the Pigeonhole Principle, show that there must be two distinct selected numbers whose quotient is a power of two. You should clearly describe what your pigeons and pigeonholes are, as well your rule for assigning pigeons to the pigeonholes.

### Answers

10 plus 100 minus 50 plus 27

Identify the sequence as arithmetic or geometric, and write a recursive formula for the sequence. Be sure to identify your starting value

The local football team won the championship several years ago, and since then, ticket prices have been increasing

$20 per year. The year they won the championship, tickets were $50. Write a recursive formula for a sequence

that models ticket prices. Is the sequence arithmetic or geometric?

### Answers

Answer:

Arithmetic Sequence

Step-by-step explanation:

Since the increase in price for each year is a constant value we can safely say that this is an arithmetic sequence. Geometric sequences have changes that are based on multiples, i.e. prices double every year or half every year are multiples of 2 and 0.5 respectively

The general formula for any arithmetic sequence is as follow

where

= Ticket price at nth year

= Starting ticket price (The year the team won the championship)

= Number of years since the championship was won

= Yearly increase in ticket price

So the formula can then be derived to be

Find an explicit form f(n) for each of the following arithmetic sequences (assume a is some real number and x is some real number).

a. −34, −22, −10, 2, ...

### Answers

Answer:

f(n) =

Step-by-step explanation:

The explicit form of an arithmetic sequence is given by the formula:

an= nth term

a1= first term

d= common difference

In this case, a1= -34

In order to obtain the value of the common difference you have to subtract two consecutive terms of the sequence (The largest minus the smallest)

-22 - (-34) = 12

To confirm that it's the common difference, subtract another two consecutive terms:

-10 - (-22) = 12

Therefore d=12

Replacing in the formula:

Identify the sequence as arithmetic or geometric, and write a recursive formula for the sequence. Be sure to identify your starting value

−101, −91, −81, −71, …

### Answers

Answer:

The sequence is arithmetic.

Step-by-step explanation:

If the sequence is arithmetic common difference will be same, if the sequence is arithmetic common ratio will be same.

Here the sequence is −101, −91, −81, −71, …

Difference between terms

-91 - (-101) = 10

-81 - (-91) = 10

-71 - (-81) = 10

They are same , so the sequence is arithmetic.

Common difference, d = 10

Now we need to find recursive formula for the sequence.

Recursive formula for GP

A radioactive substance decreases in the amount of grams by one-third each year. If the starting amount of the substance in a rock is 1,452 g, write a recursive formula for a sequence that models the amount of the substance

left after the end of each year. Is the sequence arithmetic or geometric?

### Answers

Answer:

The sequence is geometric. The recursive formula is

Step-by-step explanation:

In order to solve this problem, you have to calculate the amount of the substance left after the end of each year to obtain a sequence and then you have to determine if the sequence is arithmetic or geometric.

The substance decreases by one-third each year, therefore:

After 1 year:

Using 1452 as a common factor and solving the fraction:

You can notice that in general, after each year the amount of grams is the initial amount of the year multiplied by 2/3

After 2 years:

After 3 years:

The sequence is:

1452,968,1936/3,3872/9....

In order to determine if the sequence is geometric, you have to calculate the ratio of two consecutive terms and see if the ratio is the same for all two consecutive terms. The ratio is obtained by dividing a term by the previous term.

The sequence is arithmetic if the difference of two consecutive terms is the same for all two consecutive terms.

-Calculating the ratio:

For the first and second terms:

968/1452=2/3

For the second and third terms:

1936/3 ÷ 968 = 2/3

In conclussion, the sequence is geometric because the ratio is common.

The recursive formula of a geometric sequence is given by:

where an is the nth term, r is the common ratio and an-1 is the previous term.

In this case, r=2/3

Which of the following is equal to 4 kg

a. 4000 cg

b. 4000 g

c. 40 dag

d. 4000 mg

### Answers

4 grams is equal to 4 kg

Identify the sequence as arithmetic or geometric, and write a recursive formula for the sequence. Be sure to identify your starting value

6. 4, 40, 400, 4000, …

### Answers

Answer:

The sequence is geometric.

Step-by-step explanation:

If the sequence is arithmetic common difference will be same, if the sequence is arithmetic common ratio will be same.

Here the sequence is 4, 40, 400, 4000, …

Difference between terms

40 -4 = 36

400 - 40 = 360

They are not same , so the sequence is not arithmetic.

Ratio between terms

40/4 = 10

400/40 = 10

4000/400 = 10

They are same , so the sequence is geometric.

Now we need to find recursive formula for the sequence.

Recursive formula for GP

Please Help ASAP! Urgent! Will give Brainliest! Use the recursive formula f(n) = 0.5 ⋅ f(n − 1) + 10 to determine the 2nd term if f(1) = 4.

A. f(2) = 11.5

B. f(2) = 12.0

C. f(2) = 12.5

D. f(2) = 13.0

### Answers

Answer:

D

Step-by-step explanation:

f(n − 1) + 10

f= (4-1) = 3

3+10= 13

Identify the sequence as arithmetic or geometric, and write a recursive formula for the sequence. Be sure to identify your starting value

49, 7, 1, 1 / 7, 1 / 49, …

### Answers

Answer:

The sequence is geometric. The recursive formula is

Step-by-step explanation:

In order to identify if the sequence is arithmetic or geometric, you can calculate the difference and the ratio of two consecutive terms.

If the difference is common for all two consecutive terms,then the sequence is arithmetic. (The difference of the largest number minus the smallest one)

If the ratio is common for all two consecutive terms, then the sequence is geometric. (The ratio is obtained by dividing a term by the previous term)

Calculating the difference:

-For the first and second terms:

49-7=42

-For the second and third terms:

7-1=6

You can notice that the difference isn't common.

Calculating the ratio:

-For the first and second terms:

7÷49=1/7

-For the second and third terms:

1÷7=1/7

-For the third and fourth terms:

1/7 ÷1 =1/7

Therefore 1/7 is a common ratio and the sequence is geometric.

The recursive formula of a geometric sequence is:

where An is the nth term, An-1 is the previous term and r is the common ratio.

Replacing r=1/7:

An-1 is calculated with:

where a is the first term of the sequence.

, list the first five terms of each sequence, and identify them as arithmetic or geometric. A(n + 1) = 2 / 3 A(n) for n ≥ 1 and A(1) = 6

### Answers

Each element is 2/3 times the previous one, so that's a geometric sequence.

6×(2/3)=4, 4×(2/3)=8/3, 8/3×(2/3)=16/9, 16/9×(2/3)=32/27

Answer: 6, 4, 8/3, 16/9, 32/27

Identify the sequence as arithmetic or geometric, and write a recursive formula for the sequence. Be sure to identify your starting value

14, 21, 28, 35, …

### Answers

Answer: The sequence is an arithmetic sequence

The recursive formula is given as = + 7 , where = 14

Step-by-step explanation:

For a sequence to be geometric then there must exist a common ratio. Let r represent the common ratio. The formula for calculating common ratio implies:

r = = … , that is r is calculated by dividing the second term by the first term or the third term divided by the second term and so on.

To check if the sequence is geometric, let us find the common ratio.

= 14

= 21

= 28

= 35

So, = =

= =

= =

Considering the result, it is clear that it is not a geometric sequence since the ratios are not the same

Arithmetic Sequence , for a sequence to be arithmetic then there must be an existence of a common difference.That is

– = – = –

Let us check if the sequence given follow this rule

– = 21 -14 = 7

– = 28 – 21 = 7

– = 35 – 28 = 7

Therefore the sequence is an arithmetic sequence.

To find the recursive formula for the sequence

= 14

= +d

= +7

Every student in a discrete mathematics class is either a computer science or a mathematics major or is a joint major in these two subjects. How many students are in the class if there are 48 computer science majors (including joint majors), 23 mathematics majors (including joint majors), and 7 joint majors?

### Answers

Answer: 56 students

Step-by-step explanation:

The total number of students in the class is called the universal set

48 students are in computer science major including joint major

23 students are in Mathematics including joint major.

To find those that are in Computer science alone , subtract the joint major from the total computer science , that is

Computer science alone = 48 - 7 = 41

Also , Mathematics alone is found by subtracting joint major from the total Mathematics, that is

Mathematics alone = 23 - 7 = 16

Therefore the total number of students = Mathematics students alone + Computer science student alone + the joint majors

which is , 41 + 16 +7 =54

Therefore , there are 54 students in the class

, list the first five terms of each sequence, and identify them as arithmetic or geometric. A(n + 1) = A(n) − 19 for n ≥ 1 and A(1) = −6

### Answers

Answer:

The first 5 terms are -6,-25,-44,-63 and -82

The sequence is arithmetic

Step-by-step explanation:

Given that A(n + 1) = A(n) − 19 for n ≥ 1 and A(1) = −6

A(1) = -6

A(2) = A(1) − 19 = -6 - 19 = -25

A(3) = A(2) − 19 = -25 - 19 = -44

A(4) = A(3) − 19 = -44 - 19 = -63

A(5) = A(4) − 19 = -63 - 19 = -82

So the first 5 terms are -6,-25,-44,-63 and -82

If they are in arithmetic the common difference will be same, if geometric common ratio will be same.

Let us check common difference,

-25-(-6) = -19

-44-(-25) = -19

-63-(-44) = -19

-82-(-63) = -19

So common difference is same, so the sequence is arithmetic.