If we draw some dots on a paper, and connect each one of them to every other one with a single line, how many lines will we draw? Using mathematical induction, prove that the number of lines for n dots is 1, 2, , n(n − 1).

Step-by-step explanation:

Let us assume that we have n different dots on a paper.  We are to connect pairwise by a line.  We have to find out how many lines can be formed.

Let us prove by induction.

If there is one dot then we have no line = 1(1-1) =0

Thus n(n-1) is true for 1 dot

Let us assume that for n dots we have n(n-1) lines

Add one more point now total points are n+1.

Already the existing n points are connected by a line.

So the extra point has to be connected to each of n point

i.e. n lines should be added from the new point to the n points and again n lines from the points to the new point(Assuming lines are different if initial and final point are different)

So 2n lines would be added

So total number of lines for n+1 points

Thus true for n+1 if true for n.  Already true for n =1

So proved by induction for all natural numbers n.

197 students (rounded to the nearest whole number)

Step-by-step explanation:

first I divided 11 by 114 to find out how much girls there are to every one boy and my answer was 10.36 (rounded to the nearest 2 d.p).

next I times it by eight according to the ratio given, my answer was 83 (rounded to the nearest whole number)

finally, I just added 114 and 83 together to get the total number of students and my answer to that was 197 (rounded to the nearest whole number)

hope it helped