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Magazine Chapter 11: Problem 160
Write the first five terms of the geometric sequence, given the first term and common ratio. \(a_{1}=5, \quad r=\frac{1}{5}\)
Short Answer
Expert verified
The first five terms are 5, 1, 1/5, 1/25, 1/125.
Step by step solution
01
Understanding the Problem
We need to find the first five terms of a geometric sequence. A geometric sequence is one in which each term after the first is the product of the previous term and a fixed number called the common ratio.
02
Identifying the Formula
The formula for the nth term of a geometric sequence is given by: \( a_n = a_1 imes r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term position in the sequence.
03
Calculating the First Term
The first term \( a_1 \) is given as 5. Therefore, the first term of the sequence is 5.
04
Calculating the Second Term
Using the formula \( a_2 = a_1 imes r \), we substitute the known values: \( a_2 = 5 imes \frac{1}{5} = 1 \). So, the second term is 1.
05
Calculating the Third Term
Using the formula \( a_3 = a_1 imes r^2 \), we substitute the values: \( a_3 = 5 imes (\frac{1}{5})^2 = 5 imes \frac{1}{25} = \frac{1}{5} \). The third term is \( \frac{1}{5} \).
06
Calculating the Fourth Term
Using \( a_4 = a_1 imes r^3 \), substituting the values gives: \( a_4 = 5 imes (\frac{1}{5})^3 = 5 imes \frac{1}{125} = \frac{1}{25} \). Therefore, the fourth term is \( \frac{1}{25} \).
07
Calculating the Fifth Term
For the fifth term, \( a_5 = a_1 imes r^4 \): \( a_5 = 5 \times (\frac{1}{5})^4 = 5 \times \frac{1}{625} = \frac{1}{125} \). The fifth term is \( \frac{1}{125} \).
08
Listing the First Five Terms
The first five terms of the geometric sequence are: 5, 1, \( \frac{1}{5} \), \( \frac{1}{25} \), and \( \frac{1}{125} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Ratio
In a geometric sequence, the common ratio is a fundamental concept. It's the constant factor that we multiply each term by to get the subsequent term. In simpler words, a geometric sequence is like a multiplying pattern.
If you have a sequence where each term is achieved by multiplying the previous one with a fixed number, that fixed number is your common ratio.
For example, if the first term of a sequence is 5 and the common ratio is \(\frac{1}{5}\), it means:
If you have a sequence where each term is achieved by multiplying the previous one with a fixed number, that fixed number is your common ratio.
For example, if the first term of a sequence is 5 and the common ratio is \(\frac{1}{5}\), it means:
- To find the second term, multiply 5 by \(\frac{1}{5}\), giving you 1.
- To find the third term, take 1 and multiply it again by \(\frac{1}{5}\), giving you \(\frac{1}{5}\).
Nth Term Formula
The nth term formula of a geometric sequence is expressed as \( a_n = a_1 \times r^{(n-1)} \). This formula allows you to find any term in the sequence without having to calculate all preceding terms.
Here's a breakdown of what each element in the formula represents:
Here's a breakdown of what each element in the formula represents:
- \( a_n \) is the term you want to find.
- \( a_1 \) is the first term of the sequence.
- \( r \) is the common ratio.
- \( n \) is the position of the term in the sequence, starting with 1 for the first term.
Term Calculation
Once you understand the nth term formula, calculating the terms of a geometric sequence becomes straightforward.
Let's take a closer look at term calculation using our example:
Let's take a closer look at term calculation using our example:
- First Term: Given as 5, so there鈥檚 no need for calculation here.
- Second Term: Using the formula \( a_2 = a_1 \times r \), simply multiply 5 by \(\frac{1}{5}\) to get 1.
- Third Term: Apply \( a_3 = a_1 \times r^2 \), which translates to \(5 \times (\frac{1}{5})^2\), resulting in \(\frac{1}{5}\).
- Fourth Term: Substitute to get \( a_4 = 5 \times (\frac{1}{5})^3 = \frac{1}{25}\).
- Fifth Term: Finally compute \( a_5 = 5 \times (\frac{1}{5})^4 = \frac{1}{125}\).
Sequence Progression
Sequence progression is about understanding how a geometric sequence develops term by term.
As seen from the example sequence starting with 5, and reducing by a factor of \(\frac{1}{5}\):
As seen from the example sequence starting with 5, and reducing by a factor of \(\frac{1}{5}\):
- Each term becomes smaller as you proceed, clearly showing a decreasing pattern due to the fraction as a common ratio.
- This progression can be visualized or depicted numerically, displaying a straightforward cascading effect.
- It's easy to see how each term relates to its predecessor鈥攖hrough consistent multiplication by \(\frac{1}{5}\).
