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2 years ago

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Which of the numbers below could be terms in the sequence an=7n +4? 24 46 32 36

1 answer:

mrs_skeptik [129]2 years ago

7 0

Answer:

46

Step-by-step explanation:

46 - 4 = 42

42/7 = 6

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Prove that $5^{3^n} + 1$ is divisible by $3^{n + 1}$ for all nonnegative integers $n.$

Viktor [21]

When $n=0$ , we have

$5^{3^0} + 1 = 5^1 + 1 = 6$

$3^{0 + 1} = 3^1 = 3$

and of course 3 | 6. ("3 divides 6", in case the notation is unfamiliar.)

Suppose this is true for $n=k$ , that

$3^{k + 1} \mid 5^{3^k} + 1$

Now for $n=k+1$ , we have

$5^{3^{k+1}} + 1 = 5^{3^k \times 3} + 1 \\\\ ~~~~~~~~~~~~~ = \left(5^{3^k}\right)^3 + 1^3 \\\\ ~~~~~~~~~~~~~ = \left(5^{3^k} + 1\right) \left(\left(5^{3^k}\right)^2 - 5^{3^k} + 1\right)$

so we know the left side is at least divisible by $3^{k+1}$ by our assumption.

It remains to show that

$3 \mid \left(5^{3^k}\right)^2 - 5^{3^k} + 1$

which is easily done with Fermat's little theorem. It says

$a^p \equiv a \pmod p$

where $p$ is prime and $a$ is any integer. Then for any positive integer $x$ ,

$5^3 \equiv 5 \pmod 3 \implies (5^3)^x \equiv 5^x \pmod 3$

Furthermore,

$5^{3^k} \equiv 5^{3\times3^{k-1}} \equiv \left(5^{3^{k-1}}\right)^3 \equiv 5^{3^{k-1}} \pmod 3$

which goes all the way down to

$5^{3^k} \equiv 5 \pmod 3$

So, we find that

$\left(5^{3^k}\right)^2 - 5^{3^k} + 1 \equiv 5^2 - 5 + 1 \equiv 21 \equiv 0 \pmod3$

QED

5 0

1 year ago

A. The mean of the data is 49.

alina1380 [7]

Umm sorry i do not know that answer I am just getting the points for again so so so sorry ;(

7 0

3 years ago

Read 2 more answers

What is are if a rectangle of length of 5 and width of 3x-5y+6What is the area of a rectangle with a length of 5 and a width of

lawyer [7]

The area of rectangle is 15x - 25y + 30 sqaure units

<em><u>Solution:</u></em>

Given that, a rectangle with a length of 5 and a width of 3x - 5y + 6

To find: Area of rectangle

<em><u>The area of rectangle is given by formula:</u></em>

$Area = length \times width$

From given,

Length = 5

Width = 3x - 5y + 6

<em><u>Substituting the values in formula, we get,</u></em>

$Area = 5 \times (3x - 5y + 6)\\\\\text{Multiply 5 with each term inside the bracket }\\\\Area = 5(3x) -(5)(5y) +5(6)\\\\\text{Simplify the above expression }\\\\Area = 15x -25y + 30$

Thus area of rectangle is 15x - 25y + 30 sqaure units

6 0

3 years ago

Divide. Express your answer in simplest form.<br><br> 11/12 ÷ 3/4=

Tanzania [10]

Answer:

11/9

hope this helped :)

Step-by-step explanation:

4 0

3 years ago

Read 2 more answers

Find the difference between8/15 and 2/3 Show all calculations in your final answer.

kherson [118]

Answer:

$\frac{8}{15}-\left(\frac{-2}{3}\right)=\frac{6}{5}$

Step-by-step explanation:

Let the value of a number 'a' be = 8/15

Let the value of a number 'b' be = 2/3

The difference between the two numbers can be calculated by subtracting the numbers

$a-b=\frac{8}{15}-\left(\frac{-2}{3}\right)$

$\mathrm{Remove\:parentheses}:\quad \left(a\right)=a$

$=\frac{8}{15}-\frac{-2}{3}$

$\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{-a}{b}=-\frac{a}{b}$

$=\frac{8}{15}-\left(-\frac{2}{3}\right)$

$\mathrm{Apply\:rule}\:-\left(-a\right)=a$

$=\frac{8}{15}+\frac{2}{3}$

$\mathrm{Since\:the\:denominators\:are\:equal,\:combine\:the\:fractions}:\quad \frac{a}{c}\pm \frac{b}{c}=\frac{a\pm \:b}{c}$

$=\frac{8+10}{15}$

$=\frac{18}{15}$

$\mathrm{Cancel\:the\:common\:factor:}\:3$

$=\frac{6}{5}$

Thus,

$\frac{8}{15}-\left(\frac{-2}{3}\right)=\frac{6}{5}$

3 0

2 years ago