Use Induction to prove 2 n2 n for all n ZSolutionTo prove

Use Induction to prove 2 | n^2 + n for all n Z^+.

To prove that 2/ n² +n by induction

step 1) to prove this for n=1

cinsider n² +n = 1² +1

= 1+1

and 2/2

therefore for n=1 give statement is true

step 2) Induction hypothysis

Aussum that statementb is for n=k

i.e 2/k² +k

i.e 2/k(k+1)

step 3) to prove statement for n=k+1

consider

n² +n = (k+1)² +(k+1)

= (k+1)(k+1)+(k+1)

=(k+1)[(k+1)+1] comman k+1

=(k+1)(k+1+1)

=(k+1)(k+2)

=(k+2)(k+1)

n² +n =k(k+1)+2(k+1)

since 2/k(k+1) by hypothysis

and 2/2(k+1) (because 2(k+1)is multiple of 2)

therfor 2 divid (k+1[k)+2(k+1)] i.e

2/(k+1[k)+2(k+1)]

therefor 2/n² +n for n=k+1

therefor by induction hypothysis

2/n² +n for all n in z⁺