Finding if this recursively defined sequence satisfies this explicit formula

Hey Guys.

Having a bit of trouble with recursion and seeing if the sequence satisfies the formula.

I need to determine whether

a_{k}=2_{a}_{k-1}+k-1 for all integers k>=2

satisfies the explicit formula a_{n}=(n-1)^{2 }for all integers n>=1

Not quite sure on what the right steps are.

Thanks for your help in advance!

Re: Finding if this recursively defined sequence satisfies this explicit formula

Hey RadMabbit.

What does the 2_(ak-1) mean? Is this 2 * a_(k-1)?

Re: Finding if this recursively defined sequence satisfies this explicit formula

Sorry, typo

it is 2*a_{k-1}

so 2 multiplied by the previous a term.

Re: Finding if this recursively defined sequence satisfies this explicit formula

This is just a substitution where if a_k = (k-1)^2 then a_(k-1) = ((k-1) - 1)^2 = (k-2)^2.

Now its just a matter of checking whether LHS = RHS with that substitution.

Re: Finding if this recursively defined sequence satisfies this explicit formula

would you care to elaborate. ;)

Re: Finding if this recursively defined sequence satisfies this explicit formula

So a_k = (k-1)^2, a_(k-1) = (k-2)^2

So 2*a_(k-1) + (k-1)

= 2*(k-2)^2 + k - 1

= 2k^2 - 8k + 8 + k - 1

= 2k^2 - 7k + 7

!= (k-1)^2 = k^2 - 2k + 1 in general.

To check when they are equal equate the two and you get the condition:

k^2 - 5k + 6 = 0 or (k-2)(k-3) = 0 so if k = 2 or k = 3 then the equality holds but other-wise it doesn't.

Re: Finding if this recursively defined sequence satisfies this explicit formula

$\displaystyle a_k=1-2 k+k^2$