Verify that the sequence given by the closed formula an = n2 + n satisfies the recursive formulaan =an?1+2n.

Question

an = n2 + n satisfies the recursive formulaan an=n-1+2n.

Answer 1

Seems like a good time to use induction.  

Start with the base case n = 0, here a_0 = 0^2+0=0

And we can define A_0=0(a recursive definition always needs a starting point and using a capital A to distinguish the two definitions).  

Our base case is done, so suppose we know that for n=k, A_k=a_k, that is:

A_(k-1)+2k = k^2+k,

 

Now consider (k+1)^2+(k+1)

= k^2 + 2k + 1 + k + 1

= k^2 + 3k + 2

and

A_(k+1) = A_k+ 2(k+1)

By the induction hypothesis:

A_(k+1)= k^2+k+2(k+1)

= k^2+3k+2

Which matches the closed form expression for a_n with n = k+1.  We have shown then that if a_n and A_n match for k, they match for k+1, and by induction they must always be equal.