Define the sequence An  as:     
       n, if n is odd
An =   0, if n = 4t-2 
       1, if n = 4t
whenever t is a positive integer.

Derive a one-line expression for each of the following:

(i) A_n in terms of n

(ii) S_n in terms of n, where S_n denotes the sum of all the terms of A_n

 

The n-th term: 

 

a(n)=n* KD(n mod2,1) + KD(n mod4,0) 

 

**for definition of KD(Kronecker delta) see:

  http://reference.wolfram.com/mathematica/ref/KroneckerDelta.html 

 

Checking: 

 

a(17)=17 

a(18)=0 

a(19)=19 

a(20)=1

 

And the sum of n terms is: 

S(n)=Sum(a(1) to a(n)))= INT((N+1)/2)^2+ INT(N/4) <br>

 Checking: S(11)=6^2+2=38

 1+0+3+1+5+0+7+1+9+0+11=38

Edited on September 20, 2013, 3:31 pm

Posted by Ady TZIDON on 2013-09-20 15:27:22