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Magazine Chapter 12: Problem 8
Write the first five terms of the sequence whose general term is given. \(a_{n}=2^{n}-3 n\)
Short Answer
Expert verified
The first five terms are: -1, -2, -1, 4, 17.
Step by step solution
01
Understand the general term
The general term of the sequence is given by the formula: \[a_{n} = 2^{n} - 3n\]This means for each term, substitute the term number (n) into this equation to find the value of the term.
02
Calculate the first term (\(n = 1\))
Substitute \(n = 1\) into the formula: \[a_{1} = 2^{1} - 3(1) = 2 - 3 = -1\]
03
Calculate the second term (\(n = 2\))
Substitute \(n = 2\) into the formula: \[a_{2} = 2^{2} - 3(2) = 4 - 6 = -2\]
04
Calculate the third term (\(n = 3\))
Substitute \(n = 3\) into the formula: \[a_{3} = 2^{3} - 3(3) = 8 - 9 = -1\]
05
Calculate the fourth term (\(n = 4\))
Substitute \(n = 4\) into the formula: \[a_{4} = 2^{4} - 3(4) = 16 - 12 = 4\]
06
Calculate the fifth term (\(n = 5\))
Substitute \(n = 5\) into the formula: \[a_{5} = 2^{5} - 3(5) = 32 - 15 = 17\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
general term
In any sequence, the **general term** ( a _{n}) is a formula that determines the value of each term in the sequence using its position number ( n ). For this exercise, the general term is given as a _{n} = 2^{n} - 3n. This formula combines elements of both exponential and linear functions. Understanding the general term helps you to generate the sequence quickly.
Some tips to understand general terms better:
Some tips to understand general terms better:
- Always identify the formula: This is the backbone of your sequence calculations.
- Figure out the variables involved: Here, it includes n (the position in the sequence).
- Break down the formula: Look at the different parts, such as the exponential part ( 2 ^{n} ) and the linear part ( - 3n ).
sequence calculation
To find the first few terms of a sequence, you will need to perform **sequence calculation**. This involves substituting the position numbers into the general term formula. Let's calculate the first five terms of our sequence:
**1. First Term**:
For n = 1:
a _{1} = 2^{1} - 3(1) = 2 - 3 = -1
**2. Second Term**:
For n = 2:
a _{2} = 2^{2} - 3(2) = 4 - 6 = -2
**3. Third Term**:
For n = 3:
a _{3} = 2^{3} - 3(3) = 8 - 9 = -1
**4. Fourth Term**:
For n = 4:
a _{4} = 2^{4} - 3(4) = 16 - 12 = 4
**5. Fifth Term**:
For n = 5:
a _{5} = 2^{5} - 3(5) = 32 - 15 = 17
By substituting the respective values of n, we simplify the operations, verifying our steps correctly.
**1. First Term**:
For n = 1:
a _{1} = 2^{1} - 3(1) = 2 - 3 = -1
**2. Second Term**:
For n = 2:
a _{2} = 2^{2} - 3(2) = 4 - 6 = -2
**3. Third Term**:
For n = 3:
a _{3} = 2^{3} - 3(3) = 8 - 9 = -1
**4. Fourth Term**:
For n = 4:
a _{4} = 2^{4} - 3(4) = 16 - 12 = 4
**5. Fifth Term**:
For n = 5:
a _{5} = 2^{5} - 3(5) = 32 - 15 = 17
By substituting the respective values of n, we simplify the operations, verifying our steps correctly.
substitution method
The **substitution method** is essential when working with sequences. It involves replacing variables in a formula with specific values to calculate the outcome. For example, to find a _{n} for different n, you replace n with 1, 2, 3, etc., in 2^{n} - 3n.
Steps for using substitution method:
Steps for using substitution method:
- Identify the variable(s) to substitute.
- Replace the variable(s) with specific values.
- Perform arithmetic operations for simplification.
- Check for consistency by verifying other terms, if needed.
exponential functions
Exponential functions are a critical component of our sequence's general term. In the term a _{n} = 2^{n} - 3n, 2^{n} represents an exponential function. These functions involve raising a constant (base) to the power of an exponent (n).
Key characteristics of exponential functions:
Key characteristics of exponential functions:
- The base must be a constant (in this case, 2).
- The exponent is a variable (in this case, n).
- Exponential growth: The value rapidly increases as n increases.
linear functions
Linear functions also play a role in our sequence. In the formula a _{n} = 2^{n} - 3n, the term - 3n represents a linear function. Linear functions showcase a constant rate of change. They can be expressed in the form mx + b where m is the slope and b is the y-intercept.
Highlights of linear functions:
Highlights of linear functions:
- They increase or decrease at a constant rate.
- They produce straight-line graphs.
- The term -3n makes a consistent subtraction to every term in the sequence, providing a balancing effect against exponential growth.
