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3 years ago

Help me with this one as fast as u can

Mathematics

1 answer:

Nuetrik [128]3 years ago

6 0

Answer:

D. negative association

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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

2 years ago

I need help on this ASAP

klemol [59]

Answer:

The answer is C.

Step-by-step explanation:

When you add the same negative and positive number it's 0.

For Example:

-5 + 5= 0

6 0

2 years ago

Identify the nonlinear equation. A) y = x B) y = 3x - 7 C) y = x2 D) y = 3 E) y = 1 3

Softa [21]

ANSWER

$y = {x}^{2}$

EXPLANATION

A linear function is a function whose graph is a straight line.

The graph of y=x is a straight line.

The graph of y=3x -7 is also a straight line.

The graph of y=x² is a parabola.

This graph is not a straight line .

It is a non-linear function.

y=3 and y=13 are also straight lines.

The correct choice is C.

7 0

3 years ago

24 1/2% is equal to which

Lorico [155]

Answer:0.245

Step-by-step explanation:

5 0

3 years ago

The radius of a circle is 12.25 centimeters. What is the circumference of the circle?

Viktor [21]

With a given radius you can multiply by 2 and simply get the diameter. Getting the diameter of the circle would then allow you to use the equation pi(24.5). You can round pi to 3.14. 3.14*24.5=76.93. The circumference is 76.93.

7 0

4 years ago