We know that Un+1=3Un−2Un−1 ...........(i) Step I Given, U1=3=2+1=21+1, which is true for n=1, put n=1 in Eq.(i), Then U1+1=3U1−2U1−1 ⇒ U2=2U1−2U0 =3×3−2×2=5=22+1 Which is true for n=2 ∴ The result are true for n=1 and n=2 Step II Assume it is true for n=k, then it is also true for n=k−1 Then, Uk=2k+1 ...........(ii) and Uk−1=2k−1+1 ......(iii) Step III On putting n=k in Eq.(i), we get Uk+1=3Uk−2Uk−1 =3(2k+1)−2(2k−1+1) [From Eqs. (ii) and (iii) =3.2k+3−2.2k−1−2 =3.2+3−2k−2 =(3−1)2k+1 =2.2k+1=2k+1+1 This shows that the result is ture for n=k+1. Hence by the principle of mathematical induction the result is true for all n∊N.