Series

Question 1

# _define_:
sequence

Answer 1

An

• *ordered list** of
• *numbers**

Question 2

# _define_:
explicit formula

Answer 2

A
sequence is defined
explicitly if it
allows direct computation for
any term in the sequence

No prior term is needed

e.g.:
a_n = 4 + 3(n − 1)

Question 3

# _define_:
recursive formula

Answer 3

A
sequence is defined
recursively if the
first term is given and
there’s a way to
determine the nth term
using the
term that precedes it

The prior term is needed

e.g.:
a₁ = 4
a_n = a_n−1 + 3

Question 4

# _define_:
arithmetic sequence

Answer 4

A

• *sequence** with a
• *common difference**

e.g.:
{1, 4, 7, 10}

Question 5

# _define_:
geometric sequence

Answer 5

A

• *sequence** with a
• *common ratio**

e.g.:
{1, 2, 4, 8, 16, 32, 64 . . .}

Question 6

What is the
explicit formula
for an
arithmetic sequence?

Answer 6

a_n = a + d(n − 1)

a<sub>n</sub> = *n*th term of the sequence
a = first term of the sequence
n = term index
d = common difference

Question 7

What is the
explicit formula
for a
geometric sequence?

Answer 7

a_n = arⁿ⁻¹

a<sub>n</sub> = *n*th term of the sequence
a = first term of the sequence
​n = term index
r = common ratio

Question 8

Given
arithmetic series

{a, b, c, d},

how do you
determine the
common difference?

Answer 8

Any term
minus the
previous term

b − a
or
c − b
or
d − c

Question 9

Given
geometric series

{a, b, c, d},

how do you
determine the
common ratio?

Answer 9

Any term
divided by the
previous term

Question 10

What is an
explicit formula
for the following
sequence?

{10, 13, 16, 19, …}

Answer 10

Question 11

What is a
recursive formula
for the following
sequence?

{10, 13, 16, 19, …}

Answer 11

Question 12

What is an
explicit formula
for the following
sequence?

{3, 6, 12, 24, 48, …}

Answer 12

Question 13

What is a
recursive formula
for the following
sequence?

{3, 6, 12, 24, 48, …}

Answer 13

Question 14

What does
summation notation
do?

Answer 14

Instructs to

• *sum** the
• *elements** of a
• *sequence**

Question 15

How do you

• *write**
• *summation notation**?

Answer 15

Σ: summation sign (sigma)

• i:*index of summation
  1: starting point (lower limit of summation)
• n:*stopping point (upper limit of summation)
• a_i:*argument

Question 16

How many
terms
are there in this
summation?

Answer 16

Nine

Start at 1; stop at 9

First term: 1

Remaining terms: 2, 3, 4, 5, 6, 7, 8, 9

Total: Nine

Question 17

How many
terms
are there in this
summation?

Answer 17

Ten

Start at 0; stop at 9

First term: 0

Remaining terms: 1, 2, 3, 4, 5, 6, 7, 8, 9

Total: Ten

Question 18

What is the
formula
for a
geometric series?

Answer 18

S_n = a • (1 − rⁿ)
(1 − r)

Question 19

What is the
formula
for an
arithmetic series?

Answer 19

S_n = n • (a₁ + a_n)
2

The
average of the first and last terms
times the
number of terms