Prove these scenarios by mathematical induction:
(1) Prove n2 < 2n for all integers
n>4
(2) Prove that a finite set with n elements has 2n
subsets
(3) Prove that every amount of postage of 12 cents or more can
be formed using just 4-cent and 5-cent stamps
Prove using mathematical induction: 3.If n is a counting number
then 6 divides n^3 - n. 4.The sum of any three consecutive perfect
cubes is divisible by 9. 5.The sum of the first n perfect squares
is: n(n +1)(2n +1)/ 6
1a. Proof by induction: For every positive integer
n,
1•3•5...(2n-1)=(2n)!/(2n•n!). Please explain what the exclamation
mark means. Thank you for your help!
1b. Proof by induction: For each integer n>=8,
there are nonnegative integers a and b such that n=3a+5b
a. Use mathematical induction to prove that for any positive
integer ?, 3 divide ?^3 + 2?
(leaving no remainder).
Hint: you may want to use the formula: (? + ?)^3= ?^3 + 3?^2 * b +
3??^2 + ?^3.
b. Use strong induction to prove that any positive integer ? (? ≥
2) can be written as a
product of primes.