Sequences & Series

Question 1

What’s a sequence?

Answer 1

A function whose domain is a subset of natrual numbers. The range values are called terms of the sequence (t (subscript n))

Question 2

What does n represent in a term?

Answer 2

The term number/position

Question 3

Describe an arithmetic sequence

Answer 3

a, a + d, a + 2d

a is the first term, d is the common difference

Question 4

How do you find the common difference using terms?

Answer 4

d = tn - t(n-1)

Question 5

What is the general term of an arithmetic seq?

Answer 5

tn = a + (n - 1)d, where n is the number of terms

Question 6

What’s the formula for the mean of an arithmetic seq?

Answer 6

Mean = (a + b) / 2

Question 7

Describe a geometric sequence

Answer 7

a, ar, ar^2

a is the first term, r is the common ratio

Question 8

How do you find the common ratio using terms?

Answer 8

r = tn / t(n-1)

Question 9

What is the general term of a geometric seq?

Answer 9

tn = ar^(n-1)

Question 10

What’s the formula for the mean of a geometric seq?

Answer 10

mean = sqrt(ab)

Question 11

What is a recursive formula?

Answer 11

A formula that depends on what comes before to define a term

Question 12

What is the explicit formula?

Answer 12

A formula used to directly find any term

Question 13

What’s a series?

Answer 13

A series is the sum of all terms in a sequence

Question 14

What’s the general formula for a series sum?

Answer 14

Sn = t1 = t2 + — + tn

Useful modification: S(n-1) = t1 + t2 + — = t(n-1).
tn is left out

Question 15

How can you find the value of a term using 2 different sum formulas?

Answer 15

tn = Sn - S(n-1)

Question 16

What are the 2 formulas for the sum of an arithmetic series?

Answer 16

Sn = (n/2)(a + tn)
Sn = (n/2)(2a + (n - 1)d)

Question 17

What are the 2 formulas for the sum of a geometric series?

Answer 17

Sn = a(1 - r^n) divided by 1 - r
Sn = a(r^n - 1) divided by r - 1

In both cases, r is not equal to 0 or 1

Question 18

What’s the formula for an infinite geometric series and when do you have this case?

Answer 18

S(infinity) = a / (1-r)
It occurs when |r| <= 1, so -1 < r < 1
n approaches infinity

Question 19

What are the main applications of series?

Answer 19

Simple interest (arithmetic series) and compound interest (geometric series)

Question 20

What’s the formula for simple interest?

Answer 20

I = P x r x t

Question 21

What’s the formula for compound interest?

Answer 21

A = P(1 + i/n)^nt

Question 22

How do you solve problems w/ deposits and withdrawals?

Answer 22

Idk

Question 23

What is sigma notation?

Answer 23

A concise way to write the sum of many terms in a sequence. The nth partial sum of a series is the sum of the 1st n terms of that series

Question 24

What is the upper limit, lower limit, and general term in sigma notation?

Answer 24

The upper limit (n) is the ending term number, the lower limit (i) is the starting term number, the general term corresponds to the index i

Question 25

What are the 4 properties of sigma?

Answer 25

Review in notes