Show the following true via Mathematical InductionSolutionle

Show the following true via Mathematical Induction

let 1/1*2 + 1/2*3 +1/3*4+....+1n*(n+1)=n/n+1 ............(1)

proof:

step-1:

for n=1 equation (1) is true since

L.H.S.=1/1*2=1/2

R.H.S.=n/n+1=1/1+1=1/2

therfore L.H.S=R.H.S. ,EQUATION (1) IS TRUE for n=1

STEP(2):

suppose equation (1) is true for some n=k1 that is

1/1*2 + 1/2*3 + 1/3*4 +....+1/k*(k+1) =k/k+1 ...........(2)

STEP-3:

Prove that equation(1) is true for n=k+1, that is

1/1*2 + 1/2*3 + 1/3*4 +....+1/(k+1)*k+ 1/(k+1)*(k+2)=k+1/k+2 ......(3)

we have equation (2) i.e.1/1*2+1/2*3+1/3*4+.....+1/k*(k+1)=k/k+1,substistute this euqtion in equation 3,

then we get

---> k/k+1 + 1/(k+1)*(k+2) = k+1/k+2

---->(k(k+2)+1)/(k+1)*(k+2)=k+1/k+2

---->((k*k)+2*k+1)/(k+1)*(k+2)=k+1/k+2

---->(k+1)^2/(k+1)*(k+2)=k+1/k+2 since (k^2)+2*k+1=(k+1)^2

----->k+1/k+2=k+1/k+2

therfore L.H.S = R.H.S. equation (3) is proved

hence equation (1) is true for all n1

hence proved..