Given The Functions $f(n) = 500$ And $g(n) = (10)^{n-1}$, Combine Them To Create A Geometric Sequence, $a_n$, And Solve For The 11th Term.A. $a_n = 500 - \left(\frac{9}{10}\right)^{n-1}; \ A_{11} \approx -499.651$B.

Feb 27, 2025 by ADMIN 216 views

**Combining Functions to Create a Geometric Sequence and Solving for the 11th Term**

Introduction

In mathematics, a geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Given two functions, $f(n) = 500$ and $g(n) = (10)^{n-1}$ , we can combine them to create a geometric sequence, $a_n$ . In this article, we will explore how to combine these functions to create a geometric sequence and solve for the 11th term.

Combining Functions to Create a Geometric Sequence

To create a geometric sequence, we need to combine the two given functions, $f(n) = 500$ and $g(n) = (10)^{n-1}$ . We can do this by subtracting the second function from the first function, which will give us a new function that represents the geometric sequence.

Let's start by writing the two functions:

$f(n) = 500$

$g(n) = (10)^{n-1}$

Now, let's subtract the second function from the first function:

$a_n = f(n) - g(n)$

$a_n = 500 - (10)^{n-1}$

However, we want to create a geometric sequence with a common ratio less than 1. To do this, we can rewrite the second function as:

$g(n) = (10)^{n-1} = \left(\frac{10}{1}\right)^{n-1} = \left(\frac{10}{1}\right)^{n-1} \cdot \left(\frac{1}{10}\right)^{n-1} = \left(\frac{10}{1} \cdot \frac{1}{10}\right)^{n-1} = 1^{n-1} = 1$

Now, let's rewrite the first function as:

$f(n) = 500 = 500 \cdot 1$

Now, let's subtract the second function from the first function:

$a_n = f(n) - g(n)$

$a_n = 500 \cdot 1 - 1$

$a_n = 500 - 1$

$a_n = 499$

However, we want to create a geometric sequence with a common ratio less than 1. To do this, we can rewrite the first function as:

$f(n) = 500 = 500 \cdot \left(\frac{10}{9}\right)^{n-1}$

Now, let's subtract the second function from the first function:

$a_n = f(n) - g(n)$

$a_n = 500 \cdot \left(\frac{10}{9}\right)^{n-1} - \left(\frac{10}{1}\right)^{n-1}$

$a_n = 500 \cdot \left(\frac{10}{9}\right)^{n-1} - 10^{n-1}$

$a_n = 500 \cdot \left(\frac{10}{9}\right)^{n-1} - \left(\frac{10}{1}\right)^{n-1}$

Q: What is a geometric sequence?

A: A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: How do we combine the two given functions, $f(n) = 500$ and $g(n) = (10)^{n-1}$ , to create a geometric sequence?

A: To create a geometric sequence, we need to subtract the second function from the first function. However, we want to create a geometric sequence with a common ratio less than 1. To do this, we can rewrite the first function as:

$f(n) = 500 = 500 \cdot \left(\frac{10}{9}\right)^{n-1}$

Now, let's subtract the second function from the first function:

$a_n = f(n) - g(n)$

$a_n = 500 \cdot \left(\frac{10}{9}\right)^{n-1} - \left(\frac{10}{1}\right)^{n-1}$

Q: How do we simplify the expression for $a_n$ ?

A: We can simplify the expression for $a_n$ by factoring out the common term:

$a_n = 500 \cdot \left(\frac{10}{9}\right)^{n-1} - \left(\frac{10}{1}\right)^{n-1}$

$a_n = 500 \cdot \left(\frac{10}{9}\right)^{n-1} - 10^{n-1}$

$a_n = 500 \cdot \left(\frac{10}{9}\right)^{n-1} - \left(\frac{10}{1}\right)^{n-1}$

$a_n = 500

Introduction

Combining Functions to Create a Geometric Sequence

Q: What is a geometric sequence?

Q: How do we combine the two given functions, f(n)=500f(n) = 500f(n)=500 and g(n)=(10)n−1g(n) = (10)^{n-1}g(n)=(10)n−1, to create a geometric sequence?

Q: How do we simplify the expression for ana_nan​?

Q: How do we combine the two given functions, $f(n) = 500$ and $g(n) = (10)^{n-1}$ , to create a geometric sequence?

Q: How do we simplify the expression for $a_n$ ?