Question

Use mathematical induction to prove the statement is true for all positive integers n.

The integer n^3 + 2n is divisible by 3 for every positive integer n.

Answer 1

First put n = 1 and find whether it is true or not.
Then put n = k and assume it is true.
Find f(k+1) and prove the required thing.

Answer 2

I did n = 1, but do I have to go further than that or just n = 1?

Answer 3

You've to move on to step 2nd.

Answer 4

Say, \(f(n) = n^3 + 2n\)

i) Step - 1 :

\(f(1) = 1^3 + 2(1) = ? \)

Find whether the value "?" is divisible by 3 or not.

ii) Step - 2 :

Assume that the statement holds true for n = k. That is, f(k) is true.
That is,

\(f(k) = k^3 + 2k = 3m \) where m is any integer.

Answer 5

After this, you will find f(k+1) :

\(f(k+1) = (k+1)^3 + 2(k+1) \)

Use identities for \((a+b)^3\) and \(k^3 + 2k = 3m\) (we assumed this in Step 2nd)

Try to find f(k+1) as multiple of 3.

You will be done then.

Answer 6

oh! that seems simple enough :)

Answer 7

:) Good Luck @sleepyjess !

Answer 8

Thank you so much for helping :)