Page 185 - Discrete Mathematics and Its Applications

164 2 / Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

5 2
EXAMPLE 18 What is the value of j=1 j ?
Solution: We have
5
2
2
2
2
2
j = 1 + 2 + 3 + 4 + 5 2
j=1
= 1 + 4 + 9 + 16 + 25 ▲
= 55.
8 k
EXAMPLE 19 What is the value of k = 4 (−1) ?
Solution: We have
8
k 4 5 6 7 8
(−1) = (−1) + (−1) + (−1) + (−1) + (−1)
k = 4
= 1 + (−1) + 1 + (−1) + 1 ▲

= 1.
Sometimes it is useful to shift the index of summation in a sum. This is often done when
two sums need to be added but their indices of summation do not match. When shifting an index
of summation, it is important to make the appropriate changes in the corresponding summand.
This is illustrated by Example 20.
EXAMPLE 20 Suppose we have the sum
5
2
j
j=1
but want the index of summation to run between 0 and 4 rather than from 1 to 5. To do this,
we let k = j − 1. Then the new summation index runs from 0 (because k = 1 − 0 = 0 when
2
2
j = 1) to 4 (because k = 5 − 1 = 4 when j = 5), and the term j becomes (k + 1) . Hence,
5 4

2 2
j = (k + 1) .
j=1 k = 0
It is easily checked that both sums are 1 + 4 + 9 + 16 + 25 = 55. ▲
Sums of terms of geometric progressions commonly arise (such sums are called geometric
series). Theorem 1 gives us a formula for the sum of terms of a geometric progression.

THEOREM 1 If a and r are real numbers and r = 0, then

⎧ n+1
n ⎪ar − a
⎨ if r = 1
j

ar = r − 1
⎪
j=0 ⎩ (n + 1)a if r = 1.
Proof: Let
n
j
S n = ar .
j=0