Understanding Sequences and Patterns

The video introduces the concept of sequences, explaining them as ordered lists of numbers. It provides examples of sequences, including simple fractions and alternating patterns. The video also covers sequence notation and how to define sequences explicitly using formulas. It concludes with a preview of the next topic, which is the limit of a sequence.

What is a sequence in mathematics?

• A random collection of numbers

• An ordered list of numbers

• A set of unrelated numbers

• A single number repeated

Which of the following is an example of a sequence?

• 1, 2, 2, 1

• 1, 3, 5, 7

• 1, 1/2, 1/3, 1/4

• 1, 2, 3, 4

What pattern is observed in the sequence 1/2, 2/3, 3/4?

• Both numerator and denominator are constant

• Denominator is always 2

• Numerator increases by 1, denominator is one more than numerator

• Numerator is always 1

What is unique about the sequence of digits in pi?

• It repeats every 10 digits

• It has a predictable pattern

• It is a sequence of even numbers

• It never repeats and has no predictable pattern

What notation is used to represent sequences generally?

• a sub i

• b sub n

• c sub x

• d sub y

What is the general form of a sequence?

• a sub n = n

• a sub n = n^2

• a sub n = 1/n

• a sub n = n/(n+1)

What is the general form of a sequence?

• a sub n = n/(n+1)

• a sub n = 1/n

• a sub n = n

• a sub n = n^2

How can sequences be explicitly defined?

• By using random numbers

• By guessing the next element

• By using a formula for each element

• By listing all elements

What is the explicit formula for the sequence 1, 1/2, 1/3?

• a sub n = n-1

• a sub n = n+1

• a sub n = 1/n

• a sub n = n

What is the explicit formula for the sequence 1, 1/2, 1/3, 1/4?

• a sub n = 1/n

• a sub n = n-1

• a sub n = n

• a sub n = n+1