DiscreteMaths (computing)
- Thread starter Cairo
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i win! lol
mgoldb2
VIP Member
OK here how it works
Acording to induction you must prove 2 things
1) prove that n(1) is true
2) take N(k) and derive from it that N(K+1) is true.
Step one is easy
2(1)-1=1^2
Step two this is the hard step
You going to have to use your brain to figure out a way to do this
First you suppose N(k) is true
then we must prove N(k+1) is also true
1+....+(2(n+1))-1)=(n+1)^2
since S(k) we assume true
1+....+(2n-1)=N^2
then we can say the following
1+...+(2n-1)+(2(n+1)-1)
N^2+(2(n+1)-1) *note we said that (2n-1)=n^2 so we just making a subtitution
the rest is algebra
N^2+2n+2-1
n^2+2n+1
(n+1)^2 and now we have derive n+1 from n
there for by induction the statement holds true for all n 1 and above.
Feel free if you need me to clear anything up on how the induction works
Edit one more thing to explain why induction works
since we derive n(k+1) from n(k) then it implies that if N(1) is true then N(2) is true but N(2) implies that N(3) is true and so on for ever making induction a nice trick to prove infinite series in 2 steps only.
I would not really conisder that a proof. You need to prove that the sum equal N^2. Acutrally that the entire point of the question. Epsilon(2n-1) form i=1 to i=n is just anouther way to write the problem. It does not acturally prove anything.
Acording to induction you must prove 2 things
1) prove that n(1) is true
2) take N(k) and derive from it that N(K+1) is true.
Step one is easy
2(1)-1=1^2
Step two this is the hard step
You going to have to use your brain to figure out a way to do this
First you suppose N(k) is true
then we must prove N(k+1) is also true
1+....+(2(n+1))-1)=(n+1)^2
since S(k) we assume true
1+....+(2n-1)=N^2
then we can say the following
1+...+(2n-1)+(2(n+1)-1)
N^2+(2(n+1)-1) *note we said that (2n-1)=n^2 so we just making a subtitution
the rest is algebra
N^2+2n+2-1
n^2+2n+1
(n+1)^2 and now we have derive n+1 from n
there for by induction the statement holds true for all n 1 and above.
Feel free if you need me to clear anything up on how the induction works
Edit one more thing to explain why induction works
since we derive n(k+1) from n(k) then it implies that if N(1) is true then N(2) is true but N(2) implies that N(3) is true and so on for ever making induction a nice trick to prove infinite series in 2 steps only.
I would not really conisder that a proof. You need to prove that the sum equal N^2. Acutrally that the entire point of the question. Epsilon(2n-1) form i=1 to i=n is just anouther way to write the problem. It does not acturally prove anything.
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