1CMSC 250

Chapter 4, Summations and Products

2CMSC 250

Induction

Induction is a proof technique used to verify a property of a sequence– 2,4,6,8,… for i 1 ai = 2i

• an infinite sequence with infinite distinct values– for i 1 bi = (1)i

• an infinite sequence with finite distinct values– for 1 i 6 ci = i + 5

• a finite sequence (with finite distinct values)

3CMSC 250

Finding an explicit formula

Figure out the formula for this sequence:

,...251,

161,

91,

41,1

4CMSC 250

Finding an explicit formula Different sequences with the same initial values:

2)1(

12:0

3

kkb

kak

k

k

5CMSC 250

Summation & product notation

Sum of items specified

Product of items specified

6543216

12222222

k

k

)5(2*)4(2*)3(2*)2(2*)1(225

1

k

k

6CMSC 250

Variable ending point

n as the index of the final term

for n = 2 for n = 3

n

k knk

0

1

7CMSC 250

Nesting of sum/product notation

Variations (same or different??):

n

j

m

iij

j

Y1

2

1

)(

n

j

m

iij

j

Y1

2

1

)(

n

j

m

iij

j

Y1 1

2

8CMSC 250

Telescoping series

n

k kk

kk

1)21

1(

)1

(1

iin

i

9CMSC 250

Properties

Merging and splitting

n

mkkk

n

mkk

n

mkk baba )(

)(* * kkn

mkk

n

mkk

n

mkbaba

n

ikk

i

mkk

n

mkk aaa

1

kn

ikk

i

mkk

n

mkaaa

1*

10CMSC 250

Properties, con't.

Distribution

n

mkk

n

mkk acac )*(*

11CMSC 250

Discrete StructuresCMSC 250Lecture 22

March 24, 2008

12CMSC 250

Factorial

n! = n (n 1) (n 2) … 2 1

Definition:0! = 1n! = n (n 1)!