1) Prove that 1 + 4 + 7+....+ (3n-2) = (3n^2 - n)/2 .
2) Use induction to show that n^2 - n is always even.
Can someone help me with these problems? I have to explain this to my group so that we can present. Thanks a lot!!!
Well if you know you can avoid induction for the first part, and write it as.
Or if you prefer induction.
and i am sure you can finish that off.
the second part can be factored into the product of two consecutive numbers, so it should be fairly obvious that it must be even.
however you have been asked to do it by induction. so
suppose is even for all n up a value k
then is even. now we need to show that if it is still even
replace n with k+1 giving expand it then you add "zero" (i.e add k and then subtract k )
which is even as is even
Verify . . . True!1) Prove that:.
Add to both sides:
The left side is the left side of
The right side is: .
. .. .
This is the right side of . The inductive proof is complete.
2) Use induction to show that is always even.
Verify . . . True!
Add to both sides: .
The right side is: . ... an even number.
Add and subtract to the left side:
And we have: .
We have proved .The proof is complete.