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For the base case: n = 0.

\(\text{When }n = 0,\\ 5^{3^{0}} + 1 = 6\\ 3^{0 + 1} = 3\\ 3|6\)

Assume \(5^{3^k} + 1\text{ is divisible }3^{k + 1}\).

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

Now, \(5^{2(3^k)} \equiv 2^{2(3^k)} \equiv 4^{3^k} \equiv 1^{3^k} \equiv 1 \pmod 3\)

\(5^{3^k}\equiv 2^{3^k}\equiv 2^{2K + 1}\equiv 2 \pmod 3\) for some \(K\in \mathbb Z\)

Therefore \(3|\left(5^{2(3^k)} - 5^{3^k} + 1\right)\)

The proposition is proved.

P.S.: I cheated and used some modular arithmetic ^_^

just to clarify: HCF = Highest Common Factor?

Because this is a right triangle (10² + 24²  =  26²), the center of the circumcircle is the midpoint of the

hypotenuse. This means that the radius of the circumcircle = 26/2  =  13.

Because this is a right triangle, the radius of the incircle can be found using this formula:

   r  =  ( a + b - c ) / 2           where c is the hypotenuse and a and b are the other two sides.

From these radii, you can calculate the areas.

The triangle is a right-angled triangle.

Radius of circumcircle = 26/2 = 13

Area of circumcircle = \(\pi \cdot 13^2 = 169\pi\)

Radius of incircle = \(\dfrac{2\times\text{Area}}{\text{Perimeter}} = 4\)

Area of incircle = \(\pi \cdot 4^2 = 16\pi\)

Therefore \(a - b = 169\pi - 16\pi = 153\pi\)

a. HCF = 2x. 

\(8x^3 - 30x = 2x(4x^3 - 15)\)

b. HCF = 2

\(-2a^2 + 4x + 10 = 2(-a^2 + 2x + 5)\)

c. HCF = wx

\(7wx^2 - wx = wx(7x - 1)\)

\(\quad f(0) + f(-1) + f(-2) + f(-3)\\ = 0 + 1 + 0 + (-3)^{-2}\\ = \dfrac{10}9\)

If you can't do this by graphing, it's a painful process ...

If there are any rational solutions, the numerator is a divisor of 45:

     1, -1, 3, -3, 5. -5. 9. -9. 15. -15. 45. -45

and the denominator is a divisor of 6:

     1, -1, 2, -2, 3, -3, 6, -6

If you try enough of these, you'll find that both  -5/3  and  -3/2  are solutions.

This means that  3x + 5  and  2x + 3  are factors.

If you divide  6x4 + 49x3  128x2 + 132x + 45  by  (3x + 5)(2x + 4) 

you'll get the quotient  x2 + 5x + 3.

Then, apply the quadratic formula to  x2 + 5x + 3  to get the final two solutions.

3) for water to make the plants survive

k=9/8

To get a double root, you need the discriminant of the quadratic formula to be zero.

Discriminant:  b² - 4·a·c

Quadratic:  2x² + 3x + k     --->     a  =  2      b  =  3     c  =  k

--->     b² - 4·a·c  =  0     --->     3² - 4·2·k  =  0     --->    9 - 8k  =  0

You'll need to solve this equation for k.

there are 36+25+16+9+4+1=91 squares

Since you want  f( f(x) )  =  x, you'll want the first part to be the inverse of the second part.

Since the second part is  8 - 3x   --->     y  =  8 - 3x

(interchange both x and y)                     x  =  8 - 3y

(solve for y)                                        x - 8  =  -3y

                                                      (8 - x)/3  =  y

So:  ax + b  =  (8 - x)/3  =  8/3  - x/3  =  -(1/3)x + (8/3)

well we start with 13 and since we subtract 5 from each number we have 4-1=3 left and we do 5*3=15 finally we subtract 15 from 13 and do 13-15=-2 and so the answer is -2

You need to eliminate all the possibilities for a denominator to be zero.

Since there is a  tan(x)  term, (and since  tan(x) = sin(x)/cos(x) ), you must eliminate all the values that

make  cos(x) = 0.

Since there is a  cot(x)  term, (and since  cot(x) = cos(x)/sin(x) ), you must eliminate all the values that

make  sin(x) = 0.

Since there is a denominator of  1 + cot(x),  you must eliminate all the values that make  cot(x) = -1.

Since there is a denominator of  cot(x) - 1,  you must eliminate all the values that make  cot(x) = 1.

[And, remember that both  tan(x)  and  cot(x)  have a period of  pi.]

actually any value of \(θ\) is true if \(θ\) is a variable but the equation \({1+\tanθ \over 1 + \cotθ}\)=\({1-\tanθ \over \cotθ-1}\) is an identity  

When Carrey takes three steps, he covers 3 x 0.5 meters, or 1.5 meters.

Jim takes 4 steps for every three that Carrey takes, so he takes 4 steps to cover 1.5 meters.

1.5 meters divided by 4, gives 0.375 meters per step.

When Jim takes 24 steps, he covers 0.375 meters x 24 or 9 meters.

The two intercepts you used were not "a full wave" apart.

If you are going to use the intercept 2,4484124, you'll need to use the intercept -3.834773.

If you use a grapher, make certain that you have a sufficient amount of the graph in view, so 

that you can see what constitutes "a full wave" and get two corresponding zeros.

If you do, then you will get 2 pi.

We start with 62+55+63-32-31-38+16=95 so 95 dogs learned a trick and we do 100-95=5 so 5 dogs did not learn any tricks

Yeah, that thing happened to me it only made me have .txt ,  .html   ,   and  .zip  

I mean -4 is +(-4) sorry for making it a subtraction instead of a addition