Question

Use mathematical induction to prove that each conjecture is valid for all positive integers n.

1/3+2/3+3/3+...+n/3=n(n+1)/6

Answer 1

Explanation:

`n=1:\ (1)/(3) = (1*2)/(6)\\n=2:\ (1)/(3)+(2)/(3) = (3)/(3)=1=(2*3)/(6)=1\\n=3:\ (1)/(3)+(2)/(3)+(3)/(3) = (6)/(3)=2=(3*4)/(6)=2\\\\We\ suppose \ the\ property\ true\ for \ n :\\\\(1)/(3)+(2)/(3)+(3)/(3)+...+(n)/(3)=(n*(n+1))/(3)\\\\(1)/(3)+(2)/(3)+(3)/(3)+...+(n)/(3)+(n+1)/(3)\\\\=(n*(n+1))/(6)+(n+1)/(3)\\\\=(n+1)*((n)/(6)+(1)/(3))\\\\=(n+1)*((n+2)/(6))\\`

The property is true for n+1