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Chapter 12: Problem 9

Find a general term for each geometric sequence. $$ -5,-10,-20, \dots $$

Short Answer

Expert verified
\( a_n = -5 \times 2^{n-1} \)

Step by step solution

01

- Identify the first term

The first term of the sequence is represented as \( a_1 \). For this sequence, the first term \( a_1 \) is \( -5 \).
02

- Find the common ratio

The common ratio \( r \) can be found by dividing the second term by the first term. So, \( r = \frac{-10}{-5} = 2 \).
03

- Write the general formula for the geometric sequence

The general term of a geometric sequence is given by the formula \( a_n = a_1 \times r^{n-1} \).
04

- Substitute the known values into the formula

Using \( a_1 = -5 \) and \( r = 2 \), substitute these values into the formula: \( a_n = -5 \times 2^{n-1} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First Term
In a geometric sequence, the *first term* is the initial number you start with. This serves as the base for the entire sequence. It is commonly denoted by \( a_1 \). For example, in the sequence given: \( -5, -10, -20, \text{...} \), the first term \( a_1 \) is \( -5 \).
Common Ratio
The *common ratio* is the factor by which each term is multiplied to get the next term in a geometric sequence. It is denoted by \( r \). To find the common ratio, divide any term by the term before it. For the sequence \( -5, -10, -20, \text{...} \), the common ratio is calculated as follows:
\( r = \frac{-10}{-5} = 2 \).
This means every term is two times the previous term.
Geometric Formula
The *geometric formula* allows you to find any term in a geometric sequence. It is written as:
\[ a_n = a_1 \times r^{n-1} \]
In this formula:
  • \( a_n \) is the \text{n-th} term you want to find.
  • \( a_1 \) is the first term.
  • \( r \) is the common ratio.
Let's use this formula with our first term \( -5 \) and common ratio \( 2 \):
\( a_n = -5 \times 2^{n-1} \).
This gives you a quick way to find any term in the sequence.

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Most popular questions from this chapter

A tracer dye is injected into a system with an ingestion and an excretion. After \(1 \mathrm{hr}, \frac{2}{3}\) of the dye is left. At the end of the second hour, \(\frac{2}{3}\) of the remaining dye is left, and so on. If one unit of the dye is injected, how much is left after \(6 \mathrm{hr} ?\)

Use the formula for \(S_{n}\) to determine the sum of the terms of each geometric sequence. See \(E x-\) amples 5 and \(6 .\) In Exercises \(27-32,\) give the answer to the nearest thousandth. $$ -\frac{4}{3},-\frac{4}{9},-\frac{4}{27},-\frac{4}{81},-\frac{4}{243},-\frac{4}{729} $$

Find a general term for each geometric sequence. $$ 10,-2, \frac{2}{5}, \ldots $$

Evaluate each expression. $$ _{7} C_{4} $$

Find a general term \(a_{n}\) for the given terms of each sequence. $$ \frac{2}{5}, \frac{3}{6}, \frac{4}{7}, \frac{5}{8}, \ldots $$
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