Question

Help with mathematical induction~~

Answer 1

Prove that\[n^3 - n + 3\]is divisible by 3

Answer 2

sorry about the digression. now we do it. show that it is true if n = 1

Answer 3

\[1^3-1+3=3\] is divisible by 3

Answer 4

now assume it is true for all k less than or equal to n, and prove that under that assumption it is true for n

Answer 5

excuse me i mean it is true for n + 1

Answer 6

now some algebra
\[(n+1)^3-(n+1)+3=n^3+3n^2+2n+3=(n^3-n+3)+3n^2+3n\]

Answer 7

BY INDUCTION 3 divides
\[n^3-n+3\]and by inspection 3 divides
\[3n^2+3n\] so it must divide the whole sum and you are done

Answer 8

thanks!

Answer 9

yw