7 and 6 are parallel was that the answer you were looking for?
Answer:
use formula of equilateral triangle = roots 3/4 a square.
Answer:
D. a line and a point not on that line
Step-by-step explanation:
That is how you determine a plane.
Answer:
a) The recursion formula is
.
b)
.
Step-by-step explanation:
a) Let us explore the recurrence. A plane with only one line is divided in two regions, so
. If we add another line under the restrictions of the problem,
.
Notice that each line intersects the other n-1 lines, because there are no parallel lines. Assume we have n-1 lines and
regions in the plane. If we add a new one we will have the previous
plus n new regions. Because, for each line crossed by the new one there are a new region. Therefore,
.
b) The method here is to develop the recurrence and find some pattern. Hence, using the formula for
,
and
we obtain
![R_n = R_{n-1}+n=R_{n-2} + (n-1) + n = R_{n-3}+(n-2) + (n-1) + n.](https://tex.z-dn.net/?f=R_n%20%3D%20R_%7Bn-1%7D%2Bn%3DR_%7Bn-2%7D%20%2B%20%28n-1%29%20%2B%20n%20%3D%20R_%7Bn-3%7D%2B%28n-2%29%20%2B%20%28n-1%29%20%2B%20n.)
Notice that for each step back in the recurrence we add a new term in th sum. If we repeat the procedure n-1 times we will have
![R_n = R_{n-3}+(n-2) + (n-1) + n = R_1 + 2+3+\cdots+(n-2) + (n-1) + n.](https://tex.z-dn.net/?f=R_n%20%3D%20R_%7Bn-3%7D%2B%28n-2%29%20%2B%20%28n-1%29%20%2B%20n%20%3D%20R_1%20%2B%202%2B3%2B%5Ccdots%2B%28n-2%29%20%2B%20%28n-1%29%20%2B%20n.)
Using that
:
![R_n = R_1 + 2+3+\cdots+(n-2) + (n-1) + n = 2 +2+3+\cdots+(n-2) + (n-1) + n.](https://tex.z-dn.net/?f=R_n%20%3D%20R_1%20%2B%202%2B3%2B%5Ccdots%2B%28n-2%29%20%2B%20%28n-1%29%20%2B%20n%20%3D%202%20%2B2%2B3%2B%5Ccdots%2B%28n-2%29%20%2B%20%28n-1%29%20%2B%20n.)
Here the smart step is to split the first 2 in 1+1, in order to obtain the sum of the first n natural numbers, and the expression for this last sum it is well known. Therefore,
![R_n = 1 +(1+2+\cdots+ (n-1) + n) = 1+\frac{n(n+1)}{2}.](https://tex.z-dn.net/?f=R_n%20%3D%201%20%2B%281%2B2%2B%5Ccdots%2B%20%28n-1%29%20%2B%20n%29%20%3D%201%2B%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D.)