Using mathematical induction, the numbers a_n s are defined by a₀=1, a_ n+1 =3 n^2+n+a_n , (n 0) . Then, a_n is equal to

Step-by-step solution

Given, a₀=1, a_ n+1 =3 n^2+n+a_n array ll & a₁=3(0)^2+(0)+a₀=0+0+1=1 & a₂=3(1)^2+(1) a₁=3+1+1=5 array From option (b), Let P(n)=n^3-n^2+1 aligned & P(0)=(0)^3-(0)^2+1=1=a₀ & P(1)=(1)^3-(1)^2+1=1-1+1=a₁ & P(2)=(2)^3-(2)^2+1=8-4+1=5=a₂ aligned Thus, a_n=n^3-n^2+1