Question

Show by Mathematical Induction that the sum Sn=12+2x22+32+2x42+52+2x62 …………. is given by

Answer 1

Sn = 1² +2 *2² + 3² + 2 *4² + 5² + 2*6² + 7² + 2*8² + ...
S1 = 1, S2 = 9, S3 = 18, S4 = 50, ...

Sn = f(n) =   n(n+1)² /2 for even n.  and 
                   (n+1) n² /2 for  odd n.

PROOF BY INDUCTION:

S1 = f(1) = (1+1)*1² /2 = 1    
    The formula is true for n = 1.

1) Let n be odd and the formula apply.
    f(n) = (n+1) n² /2.
    Next term is T_{n+1} = 2 * (n+1)² , as it is the even numbered term.
    S_{n+1} = Sn + T_{n+1} = (n+1) [n² + 4n +4] /2 
                  = (n+1)(n+2)²/2 
    Same as f(n+1) = (n+1) (n+2)² /2   by applying the given formula.

2) Let n be even and formula apply.
     f(n) = n (n+1)²/2
     T_{n+1} = (n+1)².
     S_{n+1} = (n+1)² * [ n + 2 ] /2 = (n+2) (n+1)² /2 
      Same as  f(n+1) as  n+1 is odd.

So the formula is proved by Mathematical Induction.

Answer 2

S1 = f(1) = (1+1)1² /2 = 1    
    true for n = 1.

1) Let n be odd and the formula apply.
    f(n) = (n+1) n² /2.
   
    S{n+1} = Sn + T{n+1} = (n+1) [n² + 4n +4] /2 
                  = (n+1)(n+2)²/2 

2) Let n be even and formula apply.
     f(n) = n (n+1)²/2
     T{n+1} = (n+1)².
     S{n+1} = (n+1)² * [ n + 2 ] /2 = (n+2) (n+1)² /2 
      Same as  f(n+1) as  n+1 is odd.