Use mathematical induction to prove that
`n^5 + n^3 + 2n`
is divisible by 4 for integers `n >= 1.` (4 marks)
`text(See Worked Solutions)`
`text(If)\ n = 1,`
`1 + 1 + 2 xx 1 = 4\ \ text{(divisible by 4)}`
`:. text(True for)\ n = 1`
`text(Assume true for)\ n = k`
`text(i.e.)\ \ k^5 + k^3 + 2k = 4P\ \ …\ text{(1)}\ \ (text(where)\ P ∈\ text(integer))`
`text(Prove true for)\ n = k + 1`
`text(i.e.)\ \ (k + 1)^5 + (k + 1)^3 + 2(k + 1)\ \ text(is divisible by 4)`
`text(Expanding:)`
`k^5 + 5k^4 + 10k^3 + 10k^2 + 5k + 1 + k^3 + 3k^2 + 3k + 1 + 2k + 2`
`= k^5 + 5k^4 + 11k^3 + 13k^2 + 8k + 4`
`= 4P + 5k^4 + 10k^3 + 13k^2 + 8k + 4\ \ \ \ text{(see (1) above)}`
`= 4P + 4 underbrace{(k^4 + 2k^3 + 3k^2+2k + 1)}_(text(integer)\ Q) + k^4 + 2k^3 + k^2`
`= 4(P + Q) + k^2(k^2 + 2k + 1)`
`= 4(P + Q) + k^2(k + 1)^2`
`text(For any integer)\ \ k >= 2, \ k^2(k + 1)^2\ \ text{is (odd)}^2 xx text{(even)}^2`
`text{and (even)}^2 = (2R)^2\ \ text(where)\ \ R ∈\ text(integer)`
`=> k^2(k + 1)^2 = 4R^2 xx (text(odd))^2\ \ \ text{(divisible by 4)}`
`:. text(True for)\ \ n = k + 1`
`:. text(S)text(ince true for)\ \ n = 1,\ text(by PMI, true for integral)\ n >= 1.`