Name______Date______Period______Geo w/ Trig H

Introduction to Section 8.1 (Pg 537)

Part 1: Recall Prior Knowledge

1) Find the next term in each sequence.

A. 3, 9, 27, 81, ______B. 1, -5, 25, -125, ______C. 17, 11, 5, -1, -7, ______

D. 3, 5, 7, 9, 11, ______E. 64, 32, 16, 8, 4, ______F. -14, -11, -8, -5, -2, ______

2) Divide the six sequences above into two categories: (Write the letter of each sequence)

Category 1 / Category 2

You should have noticed that sequences C, D, and F use addition and subtraction to find the next term in the sequence, while sequences A, B, and E use multiplication and division to find the next term in the sequence.

Sequences C, D, and F are called arithmetic sequences. An arithmetic sequence is a numerical pattern in which each term after the first is found by adding a constant, called the common difference, d, to the previous term.

Sequences A, B, and E are called geometric sequences. A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio,r.

3) Find the mean of 18 and 56.

------Part 2: New Concepts

In #3, you found the arithmetic mean of two numbers. It is also possible to calculate the geometric mean of two numbers. Thegeometric meanof two positive numbers a and b is the POSITIVE number x such that

4) Solve for x.

From #4, you have identified the geometric mean, x, of a and b to be:

Part 3: Connect the Prior Knowledge with the New Knowledge

Look back at sequence A from Part 1. What if we knew the first and third terms only? Could we find the second term?

Let’s call the second term x. Now our sequence looks like this: 3, x, 27, … In order for this to be a geometric sequence, the ratio of the consecutive terms must be constant. So, must be the same as Therefore, It follows that and Does this “formula” look familiar?

When you find the geometric mean, x, of two numbers, a and b, you are finding the number x that makes a, x, b, be a geometric sequence.

Similarly, when you find the arithmetic mean of two numbers, you are finding a number that creates an arithmetic sequence with the two given numbers.

Let a and b be the two given numbers, and let y be their arithmetic mean. The sequence is: a, y, b, … In order for this to be an arithmetic sequence, the difference of the consecutive terms must be constant. So, y – a must be the same as b – y. Therefore, y – a = b – y. It follows that 2y = a + b and

5) Verify this connection by finding the arithmetic mean of 264 and 342. Then, write the 3 numbers out as a sequence. Is it an arithmetic sequence?

------Part 4: Apply New Knowledge

Find the geometric meanof the given numbers. If necessary, simplify your answers.

6) 3 and 25 / 7) 16 and 4
8) 9 and 13 / 9) 2 and 12

10) Challenge: If x is the geometric mean of and 2x – 9, find the value of x.