Induction Q's (1 Viewer)

[copynpaste]

Q. Examine 2^n and 2n^3 for low values of n, make a judgement which is eventually bigger, and prove your result by induction.
(For this q, I know that 2n^3 will be bigger, but not sure how to prove by induction. Also the answer is supposedly 2^n > 2n^3, for n>12... except do you show that?)

 

[HeroicPandas]

This is quick working out

Q1)
n=1, 5+2(11) = 27 which is divisible by 3

Assumption: n=k, 5^k + 2 x 11^k = 3M (where M is an integer)

Proof: n=k+1,

5^(k+1) + 2 x 11^(k+1)

= (5 x 5^k) + (2 x 11 x 11^k)

Now look at the assumption and times it by 5 -----> 5 x 5^k + 10 x 11^k = 15M ............[A]

= (5 x 5^k) + (22 x 11^k)

= (5 x 5^k) + (10 x 11^k) + (12 x 11^k)

= 15M + (12 x 11^k) (using the modified assumption .....[A])

= 3[5M + (4 x 11^k)]

= 3Q (where Q is an integer because 5M and 4 x 11^k are integers)

Therefore divisible by 3 - tada!