The Art and Craft of Problem Solving

5.3 SUMS AND PRODUCTS 157

• � d^2 = 12 + (^22) + (^52) + 102 , since dllO underneath the summation symbol means
lito

"d ranges through all divisors of 10."

f1 p^2 4 9 25 49 .. 7

• -- = -. -. -. -. .. an mfimte product
  p pnme p^2 -^1 3 8 24 48 '


• }: /(i,j) = /(3,4) + /(3,5) + /(4,5).
  3 9<j$.^5
  If the index specifications are understood in the context of a problem, they can certainly
  be omitted. In fact, often the indices get in the way of an informal, but clear, argument.
  For example,

is a reasonable, albeit technically incorrect way to write the square of a multinomial.

The precise notation is

Make sure that you understand the subscript 1 :S i < j :S n. Carefully verify (look
at examples where n = 2,3, etc.) that

� XiX j =^2 ( }: XiX j).

if.j 19 <j$.n

1 5,i,j$.n

Also verify that a summation with subscript 1 :S i < j :S n has (�) terms (you have
been reading Chapter 6, right?).

Arithmetic Series

An arithmetic sequence is a sequence of consecutive terms with a constant differ­

ence; i.e., a sequence of the form

a,a+d,a +2d, ....

An arithmetic series is a sum of an arithmetic sequence. The sum of an arithmetic

sequence is a simple application of the Gaussian pairing tool (see page 67 in Sec­

tion 3.1). Consider an arithmetic series of n terms with first term a and last term f. We

write the sum twice (d is the common difference):

S = a + (a + d) + ... + (f - d) + f,

S = f+ (f-d) + ... + (a+d) +a.

(^7) There are infinitely many primes. See Problem 2. 3. 21 on page 51 and Section 7. 1. Incidentally, the value of
this infinite product is '/t 2 /6. See Example 9 .4.8 on page^3 49.