- #1
Mouse07
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1. Prove product of 3 consecutive naturals is even data
i just want to know if this proof is acceptable, or is there a simpler proof.
Proof: suppose if the product 3 consecutive naturals is dividable by 2 then it is even.
Base case: n = 1
n (n + 1) (n + 2 )
1(2)(3)
6
since 6 is dividable by 2 base case holds
Inductive Hypo If n holds then (n+1)
inductive step
(n + 1) (n + 2)(n+3)
(n^2+3n+2)(N+3)
n^3 + 6n^2 + 11n + 6
n(n^2 + 11)+ 6 (n^2 +1)
since n (n^2 + 11) is even and 6 (n^2 + 1) is even,
and even + even = even.
since all even numbers are divisible by 2, thus n +1 hold.
hence the product of 3 consecutive naturals is even.
QED(i know that some sentences are missing but that's how i did the proof)
i just want to know if this proof is acceptable, or is there a simpler proof.
The Attempt at a Solution
Proof: suppose if the product 3 consecutive naturals is dividable by 2 then it is even.
Base case: n = 1
n (n + 1) (n + 2 )
1(2)(3)
6
since 6 is dividable by 2 base case holds
Inductive Hypo If n holds then (n+1)
inductive step
(n + 1) (n + 2)(n+3)
(n^2+3n+2)(N+3)
n^3 + 6n^2 + 11n + 6
n(n^2 + 11)+ 6 (n^2 +1)
since n (n^2 + 11) is even and 6 (n^2 + 1) is even,
and even + even = even.
since all even numbers are divisible by 2, thus n +1 hold.
hence the product of 3 consecutive naturals is even.
QED(i know that some sentences are missing but that's how i did the proof)