Sometimes, sequences are called progressions.

Arithmetic Sequence

In an Arithmetic Sequence the difference between one term and the next is a constant.

In other words, we just add the same value each time... infinitely.

In General: we could write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, ... }

where:

• a is the first term, and

• d is the difference between the terms (called the "common difference")

Rule

We can write an Arithmetic Sequence as a rule:

x_n=a+d(n-1)

(We use "n−1" because d is not used in the 1st term).

Geometric Sequence

In a Geometric Sequence each term is found by multiplying the previous term by a constant.

 In General we write a Geometric Sequence like this:

{a, ar, ar², ar³, ... }

where:

• a is the first term, and

• r is the factor between the terms (called the "common ratio")

But be careful, r should not be 0:

• When r=0, we get the sequence {a,0,0,...} which is not geometric

Rule

We can also calculate any term using the Rule:

 x_n=ar^_n-1

(We use "n-1" because ar⁰ is for the 1st term)

1)* The first term of an arithmetic sequence is 4 and the tenth term is 67.
What is the common difference?

1)* What is the eleventh term of the geometric sequence 3, 6, 12, 24, ... ?

2)** What is the thirty-second term of the arithmetic sequence -12, -7, -2, 3, ... ?

2)** What is the ninth term of the geometric sequence 81, 27, 9, 3, ... ?

3)** What is the fiftieth term of the arithmetic sequence 3, 7, 11, 15,… ?

3)*** What is the sum of the first eight terms of the geometric sequence 5, 15, 45, ... ?

4)*** What is the sum of the first sixteen terms of the arithmetic sequence 1, 5, 9, 13, ... ?

4***) What is the sum of the first nine terms of the geometric sequence 20, 10, 5, ... ? Give your answer as a decimal correct to 1 decimal place.

5)*** What is the sum of the first thirty terms of the arithmetic sequence 50, 45, 40, 35, ... ?

5)** The first term of a geometric sequence is 5 and the sixth term is 160. What is the common ratio?

6)*** What is the sum of the eleventh to twentieth terms (inclusive)of the arithmetic sequence 7, 12, 17, 22, ... ?

6)** The fourth term of a geometric sequence is 27 and the seventh term is 1. What is the first term?

7)** The fifth term of an arithmetic sequence is 11 and the tenth term is 41. What is the first term?

7)** The first term of a geometric sequence is 192 and the fifth term is 0.75. What is the common ratio?

8)*** The eleventh term of an arithmetic sequence is 30 and the sum of the first eleven terms is 55. What is the common difference?

8)*** Add up all the terms of the following infinite geometric sequence:

9)*** How many terms of the arithmetic sequence 2, 8, 14, 20, … are required to give a sum of 660?

9)*** How many terms of the geometric sequence 2, 8, 32, 128,… are required to give a sum of 174,762 ?