Induction! (1 Viewer)

[z600]

I came across this question and I just kept getting an extra term that i cant factorise.

Use mathematical induction to prove these divisibility results. Advance from k to K+2.

For even n: n^3+2n is divisible by 12

I put subbed K+2 in and i turned out an extra term (6K^2)

Could the explantion be the fact that k must be even and when u multiply is by 6 it must be a factor of 12?

thanks

 

[alcalder]

I came across this question and I just kept getting an extra term that i cant factorise.

Use mathematical induction to prove these divisibility results. Advance from k to K+2.

For even n: n^3+2n is divisible by 12

I put subbed K+2 in and i turned out an extra term (6K^2)

Could the explantion be the fact that k must be even and when u multiply is by 6 it must be a factor of 12?

thanks

For even n prove

n³+2n is divisible by 12

Test for n=2

2³+2 x 2 = 12 (which is divisible by 12)

Assume true for n=k (where k is an even interger)

Thus
k³+2k = 12P (where P is an integer)

Test k+2

(k+2)³ + 2(k+2) = k³ + 2k + 6k² + 12k + 12

But k is an even interger. Thus k = 2m (where m is an integer)

(k+2)³ + 2(k+2) = k³ + 2k + 6k² + 12k + 12

= k³ + 2k + 12mk + 12k + 12
= 12P + 12mk + 12k + 12
= 12 (P + mk + k + 1)
= 12Q (where Q is now an integer)

And then the spiel, it is proven.

 

[z600]

thanks

 

[z600]

Got 1 more

1+1/2+1/3.....1/n=Squaroot n for n>or=to7

thanks

 

[AMorris]

Taking that as meaning
1+1/2+1/3.....1/n > sqrt(n) (for all n >= 7)

1) Prove for n = 7 (plug it into your calculator and the difference is about 0.05)

2) Assume the statement true for n = k (k>=7)

3) Prove the statement true for n = k + 1:

RTP (required to prove): 1/1 + 1/2 + .. + 1/k + 1/(k+1) < sqrt(k+1)

STP (sufficient to prove): sqrt(k) + 1/(k+1) < sqrt(k + 1)

<=> 1/(k+1) < sqrt(k + 1) - sqrt(k)

... (some tedious algebra)

<=> 0 < k^2 - 2k which is true for all k>=7 (just draw the parabola)

Therefore, 1/1 + 1/2 + ... 1/n < sqrt(n) for all n >= 7 by induction.

 

[z600]

HUmm..i am stuck in the tedious algebra part, thanks anyway.

 

[AMorris]

sorry I'm a bit lazy. The basic idea is to get rid of the square roots because they impede calculations. So you put all the square roots on one side and the square it.

Now you should be left with only 1 square root. put it on its own side again and square again. Now there should be no square roots and after cancelling, the quadratic should come out (unless I made a mistake somewhere).

 

[z600]

Humm i got it but i got 6k instead of 2k.