Sequences and Series

Sequences

100

What is a sequence? Give an example.

A list of numbers in a pattern

200

1/2, -1, 3/2, -2, 5/2, -3, ...

State the pattern. Then list the next three terms in the sequence.

7/2, -4, 9/2

300

1/5, 1/10, 1/15, 1/20, ...

State the pattern. Then write a rule for the sequence.

a_n = 1/5n

Series

100

What is a series? Give an example.

The sum of the terms in a sequence

200

1 + 3 + 5 + 7 + 9

Write this sum in sigma notation. 

The sum from n=1 to 5 of 2n+1

300

How can you tell if an infinite geometric series will have a finite sum? Come up with an example of an infinite geometric series that has a finite sum. Then find its sum using the formula.

It will have a common ratio less than 1

Arithmetic Sequences/Series

100

How can you tell if a sequence is arithmetic? Give an example.

It has a common difference (something being added or subtracted to each term to get the next term)

200

State the general rule for the n^th term of an arithmetic sequence. Then come up with an example of an arithmetic sequence and find its rule.

a_n = a₁ + (n - 1)d

300

State the formula for the sum of an arithmetic sequence. Then come up with an example of an arithmetic series and find its sum using the formula.

S_n = n(a₁ + a_n)/2

Geometric Sequences/Series

100

How can you tell if a sequence is geometric? Give an example.

It has a common ratio (something being multiplied to each term to get the next term)

200

State the general rule for the n^th term of a geometric sequence. Then come up with an example of a geometric sequence and find its rule. 

a_n = a₁(r)^n-1

300

State the formula for the sum of a finite geometric sequence. Then come up with an example of a finite geometric series and find its sum using the formula.

S_n = a₁(1 - rⁿ)/(1 - r)

Recursive Sequences/Series

100

State the general rule for the n^th term of a recursive arithmetic sequence. Then give an example. 

State the general rule for the nth term of a recursive geometric sequence. Then give an example.

a_n = a_n-1 + d

a_n = a_{n-1 *} r

200

Write the first five terms (a₁ through a₅) of the sequence defined below:

a₀ = 3, a₁ = 4

a_n = a_n-1 + a_n-2

4, 7, 11, 18, 29

300

1/2, 1/4, 1/8, 1/32, ...

State the pattern. Then write a rule for the recursive sequence. Make sure to define all necessary terms.

a₁ = 1/2, a₂ = 1/4

a_n = a_n-1*a_n-2