What does the 2_(ak-1) mean? Is this 2 * a_(k-1)?
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.
Thanks for your help in advance!
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.