# Thread: Finding if this recursively defined sequence satisfies this explicit formula

1. ## 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

ak=2ak-1+k-1 for all integers k>=2

satisfies the explicit formula an=(n-1)2 for all integers n>=1

Not quite sure on what the right steps are.

2. ## Re: Finding if this recursively defined sequence satisfies this explicit formula

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

3. ## Re: Finding if this recursively defined sequence satisfies this explicit formula

Sorry, typo

it is 2*ak-1

so 2 multiplied by the previous a term.

4. ## 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.

5. ## Re: Finding if this recursively defined sequence satisfies this explicit formula

would you care to elaborate.

6. ## 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.

7. ## Re: Finding if this recursively defined sequence satisfies this explicit formula

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