B) The Ninth Term Of A Geometric Progression Is $3x$, The Seventh Term Is 16, And The Common Ratio Is Positive. Find:(i) The First Term (ii) The Sum Of The First Ten Terms
Introduction
A geometric progression 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. In this article, we will explore the concept of a geometric progression and use it to solve a problem involving the ninth and seventh terms of a geometric progression.
Understanding Geometric Progression
A geometric progression 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. The general formula for the nth term of a geometric progression is:
an = ar^(n-1)
where:
- an is the nth term of the geometric progression
- a is the first term of the geometric progression
- r is the common ratio
- n is the term number
Problem Statement
The ninth term of a geometric progression is 3x, the seventh term is 16, and the common ratio is positive. We need to find:
(i) The first term (a) (ii) The sum of the first ten terms
Solution
Step 1: Find the Common Ratio (r)
We are given that the ninth term is 3x and the seventh term is 16. We can use this information to find the common ratio (r).
Using the formula for the nth term of a geometric progression, we can write:
ar^(9-1) = 3x ar^8 = 3x
Similarly, for the seventh term:
ar^(7-1) = 16 ar^6 = 16
Now, we can divide the two equations to eliminate the first term (a):
(r^8) / (r^6) = (3x) / 16 r^2 = 3x / 16
Since the common ratio (r) is positive, we can take the square root of both sides:
r = √(3x / 16)
Step 2: Find the First Term (a)
Now that we have the common ratio (r), we can find the first term (a) using the formula for the nth term of a geometric progression.
We know that the seventh term is 16, so we can write:
ar^6 = 16
Substituting the value of r from Step 1, we get:
a(√(3x / 16))^6 = 16 a(3x / 16)^(3) = 16
Simplifying the equation, we get:
a(27x^3 / 4096) = 16
Multiplying both sides by 4096 / 27x^3, we get:
a = (16 * 4096) / (27x^3)
Simplifying the equation, we get:
a = 65536 / (27x^3)
Step 3: Find the Sum of the First Ten Terms
The sum of the first n terms of a geometric progression can be found using the formula:
S_n = a * (1 - r^n) / (1 - r)
where:
- S_n is the sum of the first n terms
- a is the first term
- r is the common ratio
- n is the number of terms
We want to find the sum of the first ten terms, so we can plug in the values:
S_10 = a * (1 - r^10) / (1 - r)
Substituting the values of a and r, we get:
S_10 = (65536 / (27x^3)) * (1 - (√(3x / 16))^10) / (1 - (√(3x / 16)))
Simplifying the equation, we get:
S_10 = (65536 / (27x^3)) * (1 - (3x / 16)^5) / (1 - (√(3x / 16)))
Conclusion
In this article, we used the concept of a geometric progression to solve a problem involving the ninth and seventh terms of a geometric progression. We found the first term (a) and the sum of the first ten terms using the formulas for the nth term and the sum of the first n terms of a geometric progression.
Final Answer
(i) The first term (a) is 65536 / (27x^3) (ii) The sum of the first ten terms (S_10) is (65536 / (27x^3)) * (1 - (√(3x / 16))^10) / (1 - (√(3x / 16)))
Introduction
In our previous article, we explored the concept of a geometric progression and used it to solve a problem involving the ninth and seventh terms of a geometric progression. In this article, we will answer some frequently asked questions about geometric progressions.
Q: What is a geometric progression?
A: A geometric progression 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: What is the formula for the nth term of a geometric progression?
A: The formula for the nth term of a geometric progression is:
an = ar^(n-1)
where:
- an is the nth term of the geometric progression
- a is the first term of the geometric progression
- r is the common ratio
- n is the term number
Q: How do I find the common ratio (r) of a geometric progression?
A: To find the common ratio (r) of a geometric progression, you can use the formula:
r = (an / a)^(1/(n-1))
where:
- an is the nth term of the geometric progression
- a is the first term of the geometric progression
- n is the term number
Q: How do I find the sum of the first n terms of a geometric progression?
A: The sum of the first n terms of a geometric progression can be found using the formula:
S_n = a * (1 - r^n) / (1 - r)
where:
- S_n is the sum of the first n terms
- a is the first term
- r is the common ratio
- n is the number of terms
Q: What is the difference between an arithmetic progression and a geometric progression?
A: An arithmetic progression is a sequence of numbers where each term after the first is found by adding a fixed number to the previous term. A geometric progression 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: Can I use a geometric progression to model real-world phenomena?
A: Yes, geometric progressions can be used to model real-world phenomena such as population growth, financial investments, and chemical reactions.
Q: How do I determine if a sequence is a geometric progression?
A: To determine if a sequence is a geometric progression, you can check if each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Q: What are some common applications of geometric progressions?
A: Geometric progressions have many common applications in finance, economics, and science. Some examples include:
- Modeling population growth
- Calculating compound interest
- Analyzing chemical reactions
- Predicting stock prices
Conclusion
In this article, we answered some frequently asked questions about geometric progressions. We hope that this article has provided you with a better understanding of geometric progressions and their applications.
Additional Resources
For more information on geometric progressions, we recommend the following resources:
- Khan Academy: Geometric Progressions
- Math Is Fun: Geometric Progressions
- Wolfram MathWorld: Geometric Progression
Note: The resources listed above are external links and are subject to change.