Use Gauss's approach to find a formula for the sum of the even numbers from 2 to 2n:
2 + 4 + 6 + ... + 2n
Your formula will be an expression involving n.

sum = n (n+1)

Step-by-step explanation:

Let S = sum

Gauss's approach

S = 2     + 4        + 6    + ...  + 2n -4  + 2n-2  +  2n

S = 2n + 2n-2 + 2n-4  +  ... +  6         + 4         + 2

add both equations term by term in order

2S = 2n+2  +  2n+2  + 2n+2 (n times

therefore

S = n(n+1)

check:

n = 1,  S = 1(1+1) = 2   [= 2]

n = 2,  S = 2(2+1) = 6   [= 2+4]

...

checks

1.4

2.3

step-by-step explanation:

the formula for the constant of variation is

y= kx

divide each side by x

y/x =kx/x

y/x =k

we know a and y

1.   y/x = k

4/1 =4

check the other values to make sure they are the same

8/2 = 4   12/3 =4   and 16/4 =4  

the constant of variation is 4

2.   y/x = k

3/1 =3

check the other values to make sure they are the same

6/2 = 3   9/3 =3   and 12/4 =3  

the constant of variation is 3

p ÷ r = tan(x)

step-by-step explanation:

given that there is a figure below shows a right triangle: a right triangle is shown with hypotenuse equal to q units and the length of the legs equal to p units and r units.

graph is missing so i will create the graph as per given information.

the angle between the legs having lengths p units and q units is y degrees. the angle between the legs having lengths r units and q units is x degrees.

now we need to find about what is p ÷ r equal to.

so we can use trigonometric ratio tan(x)=[altitude]/[base] to find the value of p ÷ r, because p is the altitude and r is the base.

p ÷ r=tan(x)

hence final answer is p ÷ r=tan(x).