Ask question

Ask question

All categories

3 years ago

7

Let n be any natural number greater than 1. Explain why the numbers n! 2, n! 3, n! 4, ..., n! n must all be composite. (This exe

rcise shows that it is possible to find arbitrarily long sequences of consecutive composite numbers.)

1 answer:

bagirrra123 [75]3 years ago

4 0

Answer:

Because each term of the sequence generates numbers with more than 1 and itself as dividers

Step-by-step explanation:

Just for the sake of correction.

$1. Explain\: why\: the\: numbers\: n! +2, n!+ 3, n!+ 4, ..., n! \\n \:must\: all\: be\: composite.$

1) Let's consider that

n! =n(n-1)(n-2)(n-3)...

2)And examine some numbers of that sequence above:

$n!+2$

Every Natural number plugged in n, and added by two will a be an even number not only divisible by two, but in some cases by other numbers for example,n=4, then 4!+2=26 which has four dividers.

3) Similarly, the same happens to

$n!+3$ and $n!+4$

Where we can find many dividers.

There's an example of a sequence, let's start with a prime number greater than 1

Let n=11

$\left \{ n!+2,n!+3,n!+4,n!+5,n!+6,...n!+n. \right \}\\\left \{ 11!+2,11!+3,11!+4,11!+5,11!+6,11!+7,11!+8,...11!+11 \right \}\\\\$

That's a long sequence of consecutive composite numbers, n=11.

$\left \{39916802, 39916803,39916804,39916805,39916806,39916807,...,39916811,39916812 \right \}$

You might be interested in

Please answer this correctly

ValentinkaMS [17]

Answer:

Step-by-step explanation:

winter 10%

spring 25%

summer 50%

fall 15%

8 0

3 years ago

Read 2 more answers

mr.browns salary is 32,000 and imcreases by $300 each year, write a sequence showing the salary for the first five years when wi

chubhunter [2.5K]

Hello!

We have the following data:

a1 (first term or first year salary) = 32000

r (ratio or annual increase) = 300

n (number of terms or each year worked)

We apply the data in the Formula of the General Term of an Arithmetic Progression, to find in sequence the salary increases until it exceeds 34700, let us see:

formula:

$a_n = a_1 + (n-1)*r$

* second year salary

$a_2 = a_1 + (2-1)*300$

$a_2 = 32000 + 1*300$

$a_2 = 32000 + 300$

$\boxed{a_2 = 32300}$

* third year salary

$a_3 = a_1 + (3-1)*300$

$a_3 = 32000 + 2*300$

$a_3 = 32000 + 600$

$\boxed{a_3 = 32600}$

* fourth year salary

$a_4 = a_1 + (4-1)*300$

$a_4 = 32000 + 3*300$

$a_4 = 32000 + 900$

$\boxed{a_4 = 32900}$

* fifth year salary

$a_5 = a_1 + (5-1)*300$

$a_5 = 32000 + 4*300$

$a_5 = 32000 + 1200$

$\boxed{a_5 = 33200}$

We note that after the first five years, Mr. Browns' salary has not yet surpassed 34700, let's see when he will exceed the value:

* sixth year salary

$a_6 = a_1 + (6-1)*300$

$a_6 = 32000 + 5*300$

$a_6 = 32000 + 1500$

$\boxed{a_6 = 33500}$

* seventh year salary

$a_7 = a_1 + (7-1)*300$

$a_7 = 32000 + 6*300$

$a_7 = 32000 + 1800$

$\boxed{a_7 = 33800}$

* eighth year salary

$a_8 = a_1 + (8-1)*300$

$a_8 = 32000 + 7*300$

$a_8 = 32000 + 2100$

$\boxed{a_8 = 34100}$

* ninth year salary

$a_9 = a_1 + (9-1)*300$

$a_9 = 32000 + 8*300$

$a_9 = 32000 + 2400$

$\boxed{a_9 = 34400}$

* tenth year salary

$a_{10} = a_1 + (10-1)*300$

$a_{10} = 32000 + 9*300$

$a_{10} = 32000 + 2700$

$\boxed{a_{10} = 34700}$

we note that in the tenth year of salary the value equals but has not yet exceeded the stipulated value, only in the eleventh year will such value be surpassed, let us see:

* eleventh year salary

$a_{11} = a_1 + (11-1)*300$

$a_{11} = 32000 + 10*300$

$a_{11} = 32000 + 3000$

$\boxed{\boxed{a_{11} = 35000}}\end{array}}\qquad\checkmark$

Respuesta:

In the eleventh year of salary he will earn more than 34700, in the case, this value will be 35000

________________________

¡Espero haberte ayudado, saludos... DexteR! =)

7 0

3 years ago

Bella is wrapping a cube-shaped gift box that measures 8.5 inches along each edge. How many square inches of wrapping paper is n

dybincka [34]

we know that

The surface area of a cube is equal to

$S=6*A$

where

A is the area of one face of the cube

in this problem

$A=8.5^{2}= 72.25\ in ^{2}$

Find the surface area

$S=6*72.25=433.5\ in ^{2}$

therefore

<u>the answer is</u>

$433.5\ in ^{2}$ of wrapping paper is needed to cover the box completely

7 0

3 years ago

77.5 as fraction or a mix number simplest form

Morgarella [4.7K]

Answer:

31/40 is a fraction of 77.5

7 0

3 years ago

Read 2 more answers

There are four activities on the critical path. Coincidentally, their standard deviations are all equal to 4. The standard devia

hichkok12 [17]

Answer:

the standard deviation of the critical path is 8

Step-by-step explanation:

The computation of the standard deviation of the critical path is shown below:

As we know that

Variance = Standard deviation^2

Now the variance of the critical path is

= Variance 1 + variance 2 + variance 3 + variance 4

= 4^2 + 4^2 + 4^2 + 4^2

= 64

Now the standard deviation would be

= √64

= 8

hence, the standard deviation of the critical path is 8

6 0

2 years ago