conman
Member
Can some one check my solution?
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Mathematical Induction (1 Viewer)
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Firstly, the question doesn't seem to make sense. If n>=2 then the equation is invalid because there is (n-3) in there. And if n=2, the number of ways is negative.conman said:Can some one check my solution?
Also, what does it mean by the "maximum number of lines joining two points"? The maximum number of lines that can join two points is exactly one in every way. Now, if this line can go through other points, that is different. But can they go through 1 other point or 2 other points or multiple number of points?
Then if n=2, there 1 way
n = 3, 2 ways
n = 4, 3 ways if you can go through 1 point, 5 ways if you can go through 2 points
Then, the proof needs to be visual with pictures. You can't just say that the number of ways of joining the points = (k-1) for n=k.
Can you reclarify the question, perhaps?
conman
Member
I've just fixed the question!!!alcalder said:
We simply assume that a max of k(k-1)/2 lines join k points.
Required to prove: a max of k(k+1)/2 lines join k+1 points (substitute n=k+1 into the original statement)
By adding (k+1)th point, we are adding k lines to the total maximum.
e.g by adding a 4th point, we are adding 3 additional lines.
. .
.
. . .
.
So for n=k+1,
total maximum lines = max no. of lines for k points + the extra number of lines added
= k(k-1)/2 + k (by assumption)
= k(k/2 - 1/2 + 1)
= k(k/2 + 1/2)
= k(k+1)/2 #
.'. true for n=k+1
Required to prove: a max of k(k+1)/2 lines join k+1 points (substitute n=k+1 into the original statement)
By adding (k+1)th point, we are adding k lines to the total maximum.
e.g by adding a 4th point, we are adding 3 additional lines.
. .
.
. . .
.
So for n=k+1,
total maximum lines = max no. of lines for k points + the extra number of lines added
= k(k-1)/2 + k (by assumption)
= k(k/2 - 1/2 + 1)
= k(k/2 + 1/2)
= k(k+1)/2 #
.'. true for n=k+1
conman
Member
Do we need to use the proof needs to be visual with pictures as alcalder mentioned above. I have checked several maths site; they suggest to use pictures as proof.