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Show that 1n^ 3 + 2n + 3n ^2 is divisible by 2 and 3 for all positive integers n.

User Charmagne
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1 Answer

4 votes

Prove:

Using mathemetical induction:

P(n) =
n^(3)+2n+3n^(2)

for n=1

P(n) =
1^(3)+2(1)+3(1)^(2) = 6

It is divisible by 2 and 3

Now, for n=k,
n > 0

P(k) =
k^(3)+2k+3k^(2)

Assuming P(k) is divisible by 2 and 3:

Now, for n=k+1:

P(k+1) =
(k+1)^(3)+2(k+1)+3(k+1)^(2)

P(k+1) =
k^(3)+3k^(2)+3k+1+2k+2+3k^(2)+6k+3

P(k+1) =
P(k)+3(k^(2)+3k+2)

Since, we assumed that P(k) is divisible by 2 and 3, therefore, P(k+1) is also

divisible by 2 and 3.

Hence, by mathematical induction, P(n) =
n^(3)+2n+3n^(2) is divisible by 2 and 3 for all positive integer n.

User Moklesur Rahman
by
7.6k points
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