# Writing Geometric Sequences Worksheets

We have learned the concept of arithmetic-based sequences earlier in middle school. They represent a data set that is maintain terms in a sense of order. Geometric sequence takes this concept to a new level where each consecutive term is multiplied a set constant. Every geometric sequence follows a rule set that can be defined by ar^{(n-1)}. The variable (a) is the first term, and the variable (r) stands for the common ratio. This rule flows because the first term is accommodated by the value of ar^{0}. These worksheets help students understand how to form and create an expression that represents a geometric sequence. You will also learn how to interpret premade sequences and determine the rule set that is present.

### Aligned Standard: HSA-SSE.B.4

- Expressions for Sequences Step-by-step Lesson- You are given a sequence where we would like you to write it as an expression.
- Guided Lesson - This type of question will scare students at first, but eventually becomes routine as the year goes on.
- Guided Lesson Explanation - I like to use reverse numbers lines to help explain this at times. The visuals help a great deal.
- Practice Worksheet - We work on this skill until you have a really good handled on what is going on with it.
- Matching Worksheet - Match the patterns to the expressions that narrate them.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

It was puzzling to see this standard at first. It has such a short scope and little real world application.

- Homework 1 - Write an equation to describe the sequence below. Use n to represent the position of a term in the sequence, where n = 1 for the first term.
- Homework 2 - Carefully plan each step of your sequence.
- Homework 3 - Find the common ratio between consecutive terms.

### Practice Worksheets

I took these sheets and made them work in a pattern to help kids along.

- Practice 1 - This is a pot of gold.
- Practice 2 - More luck of the Irish for you.
- Practice 3 - A little more work is required here.

### Math Skill Quizzes

This is quite a difficult skill to master for students.

- Quiz 1 - Write your answer using decimals and integers.
- Quiz 2 - They increase as you go.
- Quiz 3 - Place these in the right light.

### How to Write Expressions for Geometric Sequences

Understanding patterns in a numeric data set is one of the most important concepts when studying number systems. Students must grasp the concept of a variety of different sequences and series types, such as arithmetic sequences, harmonic sequences, Fibonacci numbers, and geometric sequences.

A geometric sequence is one that students are introduced to right after arithmetic sequences. In arithmetic sequences, there is a common difference "d" between two consecutive terms. The difference can either be positive or negative. In a geometric sequence, you can find the next term by multiplying a term by a common ratio "r." This multiplier cannot be equal to zero. The common term is a constant. Commonly, geometric sequences are written in the form of: [f, fr, fr^{2}, fr^{3}, …]. This is where (f) is the first term and (r) represents the common ratio. It should be noted that the three dots that trail the sequence indicates that it continues on forever. They are termed geometric sequence because as you see that increase in multiplication of the common ratio, in this case a exponents, mimics what happens in dimensional spaced geometry. A single dimension is r (linear), two dimensions is r^{2} (squared), and three dimensions is r^{3} (cubed).

To write an expression that defines a particular geometric sequence, you will have to bring into use its recursive formula which is given by; a(n)=k x r^{(n-1)}. The formula can be used to find any n^{th} term of the sequence. Here; k = first term of the sequence, r = the common ratio, n = required term number. So, how do you find the expression? Consider the following sequence; 54, 18, 6, …

**Step 1: Identify the Common Ratio** - To find the common ratio between two consecutive terms, you must divide the second term in the sequence from the first term; r = (second term)/(first term)

To check if the ratio is the same throughout, you can then divide the third term of the sequence from the second term. In this case, r is 1/3.

**Step 2: Use the Recursive Formula** - Now that you know the ratio between the terms, you can easily create an expression by inserting the terms.

a(n) = 54 [(1/3)] ^{(n-1)} The final expression can be used to find out any term in the sequence!

You might be wondering how geometric sequences are used in the real world. They have tons of uses in the financial and scientific world in situations where exponential growth is present. An example would be when you put money your bank account if it were to earn compound interest. If you place $100 in that account and it earns 3% interest annually, after the first year you would earn $3 interest. The following year you $3.09 interest because it would be based on the balance of the account ($103).