help! induction. (1 Viewer)

[hollyy.]

2007 hsc.

use mathematical induction to prove that 7^(2n+1) +5 is divisible by 12, n>=1.

urghh. step 3 is a killer.

 

[Shadose]

Step a:
Let n=1, 7^3+5= 12 . 29
->true for n=1

Step b:
assume n=k,
7^(2k+1) + 5 = 12p, is true for n>=1 where p is any integer
=7^2k . 7 + 5

Need to Prove: n=k+1
-> 7^(2k+3) +5= 12m, where m is any integer
LHS= 7^2k . 7^3 +5
=(12p-5)/7 . 7^3 +5 ,using induction hypothesis
=(12p-5) . 7^2 +5
=12p . 7^2 - 245 +5 = 12p . 7^2 - 240
=12(p . 7^2 - 20)
=12m= RHS -> it is true for n=k+1 if it is true for n=k

Step c ... annoying as always...
It is true for n=1, and it must be true for n=1+1 and so on. Therefore by mathematical induction the result is true for integers n>=1