Summation and Factorial Notations and their Properties

This set of flashcards covers key concepts and properties related to summation and factorial notations as outlined in the lecture.

What are the goals for today's discussion on summation and factorial notations?

To evaluate and simplify expressions having the summation and factorial notations, and to identify the properties of summation and factorial.

What is the definition of factorial notation?

If n! is a positive integer, then n! = n(n-1)(n-2)… We define 0! as 1.

Give an example of evaluating a factorial expression: 5!

5! = 5 · 4 · 3 · 2 · 1 = 120.

How is the sum of the first n terms of a sequence expressed in summation notation?

In summation notation, it is written as Σ{i=1}^{n} ai = a1 + a2 + … + a_n.

Summarize the first property of summation.

The first property states that Σ_{i=1}^{n} c = cn, where c is a constant.

What does the notation Σ{n i=1} (ai + b_i) equal?

It equals Σ{n i=1} ai + Σ{n i=1} bi.

What kind of problems does combinatorics help solve?

Combinatorics helps in solving counting problems, like finding the number of possible passwords.

What is the significance of counting techniques in mathematical biology?

It plays a key role in sequencing DNA.

How can the expression 1 + 1/2 + 1/3 + 1/4 + 1/5 be written using summation notation?

It can be written as Σ_{n=1}^{5} (1/n).

What is a use of summation in computing probabilities of events?

Counting techniques are used extensively when probabilities of events are computed.