1. There is no equation! You might want to chande the "-" into "="
2. Is this true for n = 1? Prove it.
3. If this is true for some k, is this true for k + 1? Write what it would mean for it to be true for k + 1, and then reduce it to the assumed truth of the statement for n = k.
You want to prove the following using the Principle of Mathematical Induction
$\displaystyle 1^2+3^2+5^2+.....+(2n-1)^2-\frac{4n^3-n}{3}=0$
Students often wonder what needs to be done on the induction step.
Suppose we had the following scenario....
The equation will be true for n=2 if it is true for n=1.
The equation will be true for n=3 if it is true for n=2.
The equation will be true for n=4 if it is true for n=3.
The equation....... on to infinity.
If the above was true for all pairs of consecutive values of n,
the equation would be true for all n>0, for n natural,
since being true for n=1 would then cause the equation to be true for all n.
We therefore try to show that if the equation is true for any n,
it will therefore be true for the next immediate n.
These 2 consecutive values of n are referred to generally as
n=k
n=k+1
Proving that the equation will be true for n=k+1 if the equation is true for n=k,
proves in general that if the equation is true for any n,
it will also be true for the next n.
Hence, write out the equation for n=k.
Write it out again with k replaced by k+1.
Try to show that if the equation is true for n=k,
then it will also be true for n=k+1.
Show that the equation is true for n=1.
i would like to say sorry and ask for forgiveness that the one i posted has an error that is instead an equation ... it should be equated by 4k^3-k / 3. this is the correct expression on the Right part of the equation to be solved by PMI... i just need your help on this guys..
Thanks a lot