prove that in N n33 n22 n6 is an integer for any n Solutio

prove that in N n^3/3 + n^2/2 + n/6 is an integer for any n

lets prove by induction.

for n=1:

expression becomes 1.

so it is an integer.

assume that statement is true for n=k.

then (k^3/3)+(k^2/2)+(k/6) is an integer.

then for n=k+1

(k^3+1+3*k^2+3*k)/3 + (k^2+2*k+1)/2 + (k+1)/6

=(k^3/3 + k^2/2 + k/6) + (k^2+2k)+ (1/3 + 1/2 + 1/6)

first bracketed item is an integer (as it is expression for n=k

(k^2+2k) is alos an integer

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

so the sum is an integer.

hence proved.