How to demonstrate this with mathematical induction?#P(n):n^3+5n# divide with 6 is true #forall ninNN#.
3 Answers
See explanation.
Explanation:
To prove this identity using mathematical induction we have to follow these steps:
- Check the identity for
#n=1# :#P(1)=1^3+5*1=6# . The result is a multiple of#6# , so the thesis is true. - Assume that it is true for
#n=k# , so:#EE{a in ZZ} k^3+5k=6a# - Next step is to prove (using the assumption) that the thesis is true for
#n=k+1#
Proof
Now we can simplify the first 2 term using the assumption from point 2.
The first component of the sum is divisible by
It is clearly divisible by
This concludes the proof.
see below
Explanation:
1) test for P(1)
2) Assume true
3) Prove for
so
but true for
See a non inductive proof.
Explanation:
To compare, a non inductive proof
We know that
so
but
and here
so