The Ninth Term Of A Geometric Progression Is 32, 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
- a is the first term
- r is the common ratio
- n is the term number
Problem Statement
The ninth term of a geometric progression is 32, the seventh term is 16, and the common ratio is positive. We need to find:
(i) The first term (ii) The sum of the first ten terms
Solution
Finding the Common Ratio
To find the common ratio, we can use the formula for the nth term of a geometric progression:
an = ar^(n-1)
We are given that the ninth term is 32 and the seventh term is 16. We can write two equations using the formula:
32 = ar^(9-1) 16 = ar^(7-1)
Simplifying the equations, we get:
32 = ar^8 16 = ar^6
Now, we can divide the two equations to eliminate the first term:
(32)/(16) = (ar8)/(ar6) 2 = r^2
Taking the square root of both sides, we get:
r = ±√2
Since the common ratio is positive, we take the positive square root:
r = √2
Finding the First Term
Now that we have the common ratio, we can find the first term using one of the equations:
32 = ar^8 32 = a(√2)^8 32 = a(2^4) 32 = a(16) a = 32/16 a = 2
So, the first term is 2.
Finding the Sum of the First Ten Terms
The sum of the first n terms of a geometric progression can be found using the formula:
Sn = a(1 - r^n)/(1 - r)
where:
- Sn 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 are asked to find the sum of the first ten terms, so we can plug in the values:
Sn = 2(1 - (√2)^10)/(1 - √2) Sn = 2(1 - 2^5)/(1 - √2) Sn = 2(1 - 32)/(1 - √2) Sn = 2(-31)/(1 - √2) Sn = -62/(1 - √2) Sn = -62/(1 - √2) × (1 + √2)/(1 + √2) Sn = -62(1 + √2)/(1 - 2) Sn = -62(1 + √2)/(-1) Sn = 62(1 + √2) Sn = 62 + 62√2
So, the sum of the first ten terms is 62 + 62√2.
Conclusion
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&A
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
- a is the first term
- r is the common ratio
- n is the term number
Q: How do I find the common ratio of a geometric progression?
A: To find the common ratio, you can use the formula for the nth term of a geometric progression and divide two consecutive terms. For example, if the nth term is an and the (n+1)th term is an+1, you can divide an+1 by an to get the common ratio:
r = an+1/an
Q: How do I find the first term of a geometric progression?
A: To find the first term, you can use the formula for the nth term of a geometric progression and substitute the values of the nth term and the common ratio. For example, if the nth term is an and the common ratio is r, you can substitute the values into the formula:
an = ar^(n-1)
and solve for a.
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:
Sn = a(1 - r^n)/(1 - r)
where:
- Sn 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 called the common difference to the previous term. A geometric progression, on the other hand, 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 a geometric progression have a negative common ratio?
A: Yes, a geometric progression can have a negative common ratio. However, if the common ratio is negative, the terms of the progression will alternate in sign.
Q: Can a geometric progression have a common ratio of 1?
A: Yes, a geometric progression can have a common ratio of 1. In this case, the terms of the progression will be equal.
Q: Can a geometric progression have a common ratio of -1?
A: Yes, a geometric progression can have a common ratio of -1. In this case, the terms of the progression will alternate in sign.
Conclusion
In this article, we answered some frequently asked questions about geometric progressions. We hope that this article has been helpful in clarifying any confusion you may have had about geometric progressions. If you have any further questions, please don't hesitate to ask.
Additional Resources
Geometric Progression Formula
The formula for the nth term of a geometric progression is:
an = ar^(n-1)
where:
- an is the nth term
- a is the first term
- r is the common ratio
- n is the term number
Geometric Progression Examples
- The sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3.
- The sequence 1, 1/2, 1/4, 1/8, ... is a geometric progression with a common ratio of 1/2.
Geometric Progression Problems
- Find the 10th term of the geometric progression 2, 6, 18, 54, ...
- Find the sum of the first 10 terms of the geometric progression 1, 1/2, 1/4, 1/8, ...