Sequences and Series

Question 1

What defines am arithmetic sequence?

Answer 1

It’s a sequence in which each term is increased by a common difference to get to the the next term.

Question 2

What defines a geometric sequence?

Answer 2

It’s a sequence in which each term is multiplied by a common ratio to get to the the next term.

Question 3

What are methods to find the common ratio, r, of a geometric sequence?

Answer 3

1. Divide two consectutive terms. (For example: u₂/u₁ or u₄/u₃.)
2. Set up a system of equations using the information you’re given.
   
   • If you know two terms or two sums:
     
     • Solve one equation for u₁
     • Substitute this value into the other equation to solve for r.
   • If you know u₁ and a sum, plug u₁ into the sum formula and solve for r.

Question 4

What are methods to find the common difference, d, of an arithmetic sequence?

Answer 4

1. Subtract two consectutive terms. (For example: u₂–u₁ or u₄ – u₃.)
2. Set up a system of equations using the information you’re given.
   
   • If you know two terms or two sums:
     
     • Solve one equation for u₁
     • Substitute this value into the other equation to solve for d.
   • If you know u₁ and a sum, plug u₁ into the sum formula and solve for r.

Question 5

Why would a question tell you that u₁, u₂, u₃, u_{<span>4</span>}, is a geometric sequence without actally giving you values or a function?

Answer 5

The intend for you to rewrite them in terms of u₁ so that you can use them and substitute them somewhere.

Question 6

Why would a question tell you that u₁, u₂, u₃, u_{<span>4</span>}, is an arithmetic sequence without actally giving you values or a function?

Answer 6

The intend for you to rewrite them in terms of u₁ so that you can use them and substitute them somewhere.

Question 7

What does S_n mean?

Answer 7

It’s the sum of a sequence of numbers.

This is also called a series.

Question 8

How can you find u_n if you are given a formula for S_n?

Answer 8

S_n is simply what you get when you add up all of the u terms.

This means that S_n=S_n-1+u_n

You can plug n and n-1 into the summation equation and solve for u_n.

(This will work no matter what type of sequence it is.)

Question 9

How do you calculate a summation