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f(n) = n^2 + 2n + 1.

4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , >>Possible sides of a triangle = 31

\(3(x+2)=20-4x\)

= \(3x + 6 = 20-4x\)

= \(3x + 6 + 4x = 20\)

= \(3x + 4x = 20 - 6\)

= \(7x = 14 \)

 = \(x = 2\)

Let's check our work

= \(3(2 + 2)=20- 4(2)\)

= \(3(4)=20-8\)

= \(12 = 12\)

X = 2

As follows:

Copy and paste it into the calculator on the Home page here and press enter.

(1, 4, 15) , (1, 5, 14) , (1, 6, 13) , (1, 7, 12) , (1, 8, 11) , (1, 9, 10) , (2, 3, 15) , (2, 4, 14) , (2, 5, 13) , (2, 6, 12) , (2, 7, 11) , (2, 8, 10) , (3, 4, 13) , (3, 5, 12) , (3, 6, 11) , (3, 7, 10) , (3, 8, 9) , (4, 5, 11) , (4, 6, 10) , (4, 7, 9) , (5, 6, 9) , (5, 7, 8) , Total =  22 such triplets

Assuming you meant (p)(q^3^3)(4)(p^2)(q^2)(2)(p)(q^2^3)

adding the exponents would give

8P⁴q¹⁷

x+5=y-5

y+14=2(x-14)

   x -y   = -10

-2x+y   = -32

x=42  

y=52

Brother has $42

Basically, we have a lattice grid. There total number of ways to get from the Origin to (4, 6) is 10 choose 4 = 210. The total number of ways to get from (0,0) to (2, 4) is 6 choose 2, and from (2, 4) to (4, 6) is 4 choose 2. 6 choose 2 * 4 choose 2 = 90. Subtracting, the total number of ways marvin can reach his home is 210 - 90 = 120 ways.

N, N+1, N+2, N+3, N+4, N+5, N+6 

N+ N+1+ N+2+ N+3+ N+4+ N+5+ N+6=6(N+6)

7N+21=6N+36

N=15

(b) The answer is (-3/5, 4/5, -12/13, 5/13)

A general term in the expansion is \(\begin{pmatrix} 9\\ n \end{pmatrix} (2z)^nz^{-(\frac{9-n}{2})}(-1)^{9-n}\)

The constant term means the z terms don't appear, which means we must have \(n-\frac{9-n}{2}=0\)  or n = 3

So the constant is given by \(\begin{pmatrix} 9\\ 3 \end{pmatrix}2^3(-1)^6=672\)

You have not incorporated the condition "... at least three of the squares are red" Guestonwebsite.

Go to https://www.desmos.com/calculator and simply type in your equation.

Well if you want to know the geeky half angle formula for contangents, here it is:

cot theta/2 = squareroot{(1 + cos theta)/(1 - cos theta)}

ta dah :D