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Magazine Chapter 12: Problem 5
Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=2^{n-1}$$
Short Answer
Expert verified
1, 2, 4, 8, 16.
Step by step solution
01
Identify the Sequence Formula
The formula for the sequence provided is \( a_n = 2^{n-1} \). This means each term is obtained by raising 2 to the power of \( n-1 \).
02
Calculate the First Term
To find the first term \( a_1 \), substitute \( n = 1 \) into the formula: \( a_1 = 2^{1-1} = 2^0 = 1 \).
03
Calculate the Second Term
To find the second term \( a_2 \), substitute \( n = 2 \) into the formula: \( a_2 = 2^{2-1} = 2^1 = 2 \).
04
Calculate the Third Term
To find the third term \( a_3 \), substitute \( n = 3 \) into the formula: \( a_3 = 2^{3-1} = 2^2 = 4 \).
05
Calculate the Fourth Term
To find the fourth term \( a_4 \), substitute \( n = 4 \) into the formula: \( a_4 = 2^{4-1} = 2^3 = 8 \).
06
Calculate the Fifth Term
To find the fifth term \( a_5 \), substitute \( n = 5 \) into the formula: \( a_5 = 2^{5-1} = 2^4 = 16 \).
07
List the First Five Terms
The first five terms of the sequence are 1, 2, 4, 8, and 16.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sequence Formula
A sequence formula is a mathematical expression that represents elements of a sequence in a logical order. In the context of arithmetic sequences, the sequence formula determines how we calculate each term. For the exercise given, the sequence formula is \( a_n = 2^{n-1} \). This indicates the term \( a_n \) is obtained by raising 2 to the power of \( n-1 \). Understanding this conceptual framework makes it much easier to calculate each term without any tools like calculators.
- Each sequence term depends on its position \( n \).
- Substitute the value of \( n \) into the formula to find the corresponding term.
- This particular sequence grows exponentially, as each subsequent term is a power of 2.
Exponents
Exponents are a crucial part of mathematics that simplify long multiplication processes. When we talk about exponents, we mean repeatedly multiplying a number by itself. In this exercise, the number 2 is being raised to the power of \( n-1 \). For example, raising 2 to the power of 3 is written as \( 2^3 \), which equals 8 because it is equivalent to multiplying 2 by itself three times (\( 2 \times 2 \times 2 = 8 \)).
Exponents have several important properties that make calculations simpler:
Exponents have several important properties that make calculations simpler:
- **Power of Zero**: Any number raised to the power of zero is 1. For example, \( 2^0 = 1 \).
- **Power of One**: Any number raised to the power of one is the number itself. Hence, \( 2^1 = 2 \).
- **Higher Powers**: As the exponent increases, the result grows rapidly due to repeated multiplication.
Term Calculation
Calculating the terms in a sequence can initially seem daunting, but with a clear process, it becomes straightforward. Let's delve into how term calculation was performed in the provided exercise. To find a term's value, \( n \) is substituted into the sequence formula \( a_n = 2^{n-1} \). This simple substitution allows us to easily calculate any term:
- **First Term**: \( n = 1 \). Thus, \( a_1 = 2^{1-1} = 2^0 = 1 \).
- **Second Term**: \( n = 2 \). So, \( a_2 = 2^{2-1} = 2^1 = 2 \).
- **Third Term**: \( n = 3 \). Therefore, \( a_3 = 2^{3-1} = 2^2 = 4 \).
- **Fourth Term**: \( n = 4 \). Thus, \( a_4 = 2^{4-1} = 2^3 = 8 \).
- **Fifth Term**: \( n = 5 \). So, \( a_5 = 2^{5-1} = 2^4 = 16 \).
