SOLUTION: Hello, could I please have some help with this question, Prove by mathematical induction that; (1 � 2) + (3 � 4) + � � � + (2n − 1) � 2n = n(n + 1)(4n − 1)/3 for

Question 1075196: Hello,
could I please have some help with this question,
Prove by mathematical induction that;
(1 � 2) + (3 � 4) + � � � + (2n − 1) � 2n = n(n + 1)(4n − 1)/3
for all n ∈ N.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The left hand side of the equation is the sum of
products of the form ,
starting from .
You need to prove that
1) it is true for , the fisrt natural number in N.
2) If it is true for , it must be true for .

1) For the equation becomes
The equation becomes
<---> ,
which is obviously true.

2) For the equation becomes
.
The sum on the left hand sides has terms,
If the formula you want to prove is true for ,
those terms add to .

For the equation becomes

and that is what you have to prove, staring from
.
The sum on the left hand sides have the same first terms
up to , the term number ,

Starting from
,
lets add to both sides of the equal sign.
.
Now the left hand side looks like the sum of terms you want.
If you get the right hand side to look like
,
you have proven part 2), so it's just a little algebra from now on.
.
.
The sum on the left hand sides has terms,
If the formula you want to prove is true for ,
those terms add to .

At this point, you realize that ,
so it's already really proven that
if the formula you had to prove works for
it must work for .
You can continue, writing
,
and
.

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SOLUTION: Hello, could I please have some help with this question, Prove by mathematical induction that; (1 � 2) + (3 � 4) + � � � + (2n &#8722; 1) � 2n = n(n + 1)(4n &#8722; 1)/3 for

SOLUTION: Hello, could I please have some help with this question, Prove by mathematical induction that; (1 � 2) + (3 � 4) + � � � + (2n − 1) � 2n = n(n + 1)(4n − 1)/3 for