 
Summary: Progressions and Sums
Putnam Practice
October 6, 2004
An arithmetic progression is a sequence of numbers {ak} with a2 
a1 = a3  a2 = ... = d. Note
ak =
ak1 + ak+1
2
and ak = a1 + (k  1)d
A sequence {bk} is a geometric progression if there is a number r
such that bk+1 = rbk. The sum of a finite geometric progression is
Sn =
b1(1  rn)
1  r
if r = 1, otherwise it is nb1.
Denote by Sr(n) = n
k=1 kr, then
S1(n) = n(n + 1)/2
S2(n) = n(n + 1)(2n + 1)/6
S3(n) = (n(n + 1)/2)2
