Question

I am not asking for a solution for my homework , i made a solution for this equation and just wanna know whether it is true

T(n) = 2 * T(n-2) - 15
T(1) = T(2) = 40

SOLUTION

Level 1
     T(n) = 2 ( T(n-4) -15 ) -15
Level 2
          = 2 ( 2 ( T(n-6) -15 ) -15 ) -15
Level 3
          = 2 ( 2 ( 2 ( T(n-8) -15 ) -15 ) -15 ) -15

From these substitutions i concluded that

 T(n) = 2^i [ T(n -2(i+1) ] - (2^(i+1) -1 ) * 15

So the solution i reached was

 T(n) = 25 * 2^[ (n-1)/2 ] -15
 AND I USED T(1) = 40

BUT the book i am reading : ” Algorithms analysis : an active learning approach ” used T(2) = 40 and reached another solution

IS MY SOLUTION TRUE TOO ?

Note I am using the direct substitution here not any other method like Master or substitution method

Thanks

Answer 1

When deriving formulae, one good way of checking is to substitute and check it with a known case.

In this case, substituting 1 results in T(1) = 25*2^((1-1)/2) - 15 = 10, but T(1) should be 40.

Additionally, in the derivation, the level 1 derivation should be 2( 2T(n-4) - 15) - 15 instead of 2( T(n-4) - 15) - 15.