Expert: Scott A Wilson - 2/25/2010

Question
I'm really stuck on how to determine the sums of:\

Sn=(x+2y)+(x+5y)+(x+8y)+(x+11y)+...

Sn= (xy^-2)+(xy^-6)+(xy^-10)+(xy^-14)+...

and

Sn= (x+y^2)+(3x+y^6)+(5x+y^10)+(7x+y^14)...


Thank you so much in advance!

Answer
For the 1st Σ, x is in every term.  The difference of the y's is 3y, so the summation would be
-ky + (x+3y)Σ(k), going from 1 to k.  It is known that Σ(k) n(n+1)/2 if k goes from 1 to n.

For the 2nd Σ, it is known that Σt^k from 1 to n is (1-t^(n+1))/(1-t).
Note that the Σ is of (x/y²)Σ(1/y^4)^k.  Thus, use t=1/y^4, assume the sum is 0 to n,
and the expression for this I just gave.

For the 3rd Σ, It can be split into x + 2xΣk + y²Σy^4, both sums from 0 to n.
The 1st sum here uses Σk = n(n+1)/2 and the 2nd sum uses (1-t^(n+1))/(1-t) where t = y^4.