F.4 Mathematical Induction

Let 5^k + 2(11^k) - 3 is divisibleby 12
when n = k+1 :
5^(k+1) + 2(11^(k+1)) - 3
= 5(5^k) + 22(11^k) - 3
= 5(5^k) + 10(11^k) - 15 + 12(11^k) + 12
= 5(5^k + 2(11^k) - 3) + 12(11^k + 1)
= .................................................
n^4+2n^3-n^2+14n is divisible by 8
Let n = k be true ,
when n = k+1 :
(k+1)^4 + 2(k+1)^3 - (k+1)^2 + 14(k+1)
=(k^4 + 4k^3 + 6k^2 + 4k + 1) + (2k^3 + 6k^2 + 6k + 2) - (k^2 + 2k + 1) + 14k + 14
= k^4 + 6k^3 + 11k^2 + 22k + 16
= (k^4+2k^3-k^2+14k) + (4k^3 + 12k^2 + 8k + 16)
= (k^4+2k^3-k^2+14k) + 4(k^3 + 3k^2 + 2k + 4)
consider 4(k^3 + 3k^2 + 2k + 4),when k is odd :
4(odd + odd + even + 4) = 4 * even is divisible by 8
when k is even :
4(even + even + even + 4) = 4*even is divisible by 8