Chapter 3 - Sequences and Series

Question 1

What is an arithmetic sequence?

Answer 1

A sequence where the differentce between each consecutive term is constant. eg. 1, 4, 7, 10

Question 2

What is the formula for the nth term of an arithmetic sequence?

Answer 2

nth term is u_n = a + (n - 1)d

where a is the 1st term and d is the difference

Question 3

What is an arithmetic series?

Answer 3

Sum of the terms of an arithmetic sequence

Question 4

What is the sum of the first n terms of an arithmetic series?

Answer 4

S_n = n/2(2a + (n-1)d)

or S_n = n/2(a + l) where l = last term

Question 5

What is a geometric sequence?

Answer 5

A sequence where there is a common ratio between consecutive terms e.g 1, 3. 9. 27

Question 6

What is the nth term of a geometric sequence?

Answer 6

u_n = ar^n-1

Question 7

What is the sum of the first n terms of a geometric series?

Answer 7

S_n = a(1 - rⁿ)

1 - r

or

S_n = a(rⁿ - 1)

r - 1

Question 8

When is a geometric series convergent?

Answer 8

When | r | <1

Question 9

What is the sum to infinity of a convergent series?

Answer 9

S (to infinity) = a / (1 - r)

Question 10

What does Σ mean?

Answer 10

Σ means sum, you write limits on the bottom and top to show which terms you are summing.

Question 11

What is a recurrent sequence?

Answer 11

A sequence of the form u_n+1 = f(u_n) where each term of a sequence is defined as a function of the previous term.

Question 12

When is a recurrent sequence increasing?

Answer 12

When u_n+1 > u_n for all n € N

Question 13

When is a recurrent sequence decreasing?

Answer 13

When u_n+1 < u_n for all n € N

Question 14

What is a perodic sequence?

Answer 14

A sequence is periodic if it repeats in a cycle so that u_n+k = u_n

The value k is called the order of the sequence.