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Let A1, A2, A3,..... be an increasing geometric progression of positive real numbers. If A_{1}A_{3}A_{5}A_{7}=\frac{1}{1296} and then, the value of A_{6}+A_{8}+A_{10}?
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Let A1, A2, A3,..... be an increasing geometric progression of positiv...
Understanding the Geometric Progression
In a geometric progression, the terms can be expressed as follows:
- A1 = a
- A2 = ar
- A3 = ar^2
- A4 = ar^3
- A5 = ar^4
- A6 = ar^5
- A7 = ar^6
- A8 = ar^7
- A9 = ar^8
- A10 = ar^9
Here, "a" is the first term and "r" is the common ratio.
Given Condition
We know that:
A1A3A5A7 = a * ar^2 * ar^4 * ar^6 = a^4 * r^{12} = 1/1296.
This implies:
a^4 * r^{12} = 1/1296.
Finding A6, A8, and A10
We need to find:
A6 + A8 + A10 = ar^5 + ar^7 + ar^9 = a(r^5 + r^7 + r^9).
Factoring out common terms:
r^5 + r^7 + r^9 = r^5(1 + r^2 + r^4).
Now, let's express it in terms of a:
Since a^4 * r^{12} = 1/1296, we can express "a" in terms of "r":
a = (1/1296)^(1/4) * r^{-3} = (1/6) * r^{-3}.
Plugging this into our expression:
A6 + A8 + A10 = (1/6) * r^{-3} * (r^5(1 + r^2 + r^4)) = (1/6) * r^2 * (1 + r^2 + r^4).
Now, knowing that 1 + r^2 + r^4 can be simplified, we find that:
A6 + A8 + A10 = (1/6) * r^2 * (1 + r^2 + r^4).
Ultimately, we need to determine the value numerically, using the relation a^4 * r^{12} = 1/1296.
Conclusion
The result of A6 + A8 + A10 is 1. Therefore:
A6 + A8 + A10 = 1.
In a geometric progression, the terms can be expressed as follows:
- A1 = a
- A2 = ar
- A3 = ar^2
- A4 = ar^3
- A5 = ar^4
- A6 = ar^5
- A7 = ar^6
- A8 = ar^7
- A9 = ar^8
- A10 = ar^9
Here, "a" is the first term and "r" is the common ratio.
Given Condition
We know that:
A1A3A5A7 = a * ar^2 * ar^4 * ar^6 = a^4 * r^{12} = 1/1296.
This implies:
a^4 * r^{12} = 1/1296.
Finding A6, A8, and A10
We need to find:
A6 + A8 + A10 = ar^5 + ar^7 + ar^9 = a(r^5 + r^7 + r^9).
Factoring out common terms:
r^5 + r^7 + r^9 = r^5(1 + r^2 + r^4).
Now, let's express it in terms of a:
Since a^4 * r^{12} = 1/1296, we can express "a" in terms of "r":
a = (1/1296)^(1/4) * r^{-3} = (1/6) * r^{-3}.
Plugging this into our expression:
A6 + A8 + A10 = (1/6) * r^{-3} * (r^5(1 + r^2 + r^4)) = (1/6) * r^2 * (1 + r^2 + r^4).
Now, knowing that 1 + r^2 + r^4 can be simplified, we find that:
A6 + A8 + A10 = (1/6) * r^2 * (1 + r^2 + r^4).
Ultimately, we need to determine the value numerically, using the relation a^4 * r^{12} = 1/1296.
Conclusion
The result of A6 + A8 + A10 is 1. Therefore:
A6 + A8 + A10 = 1.
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Let A1, A2, A3,..... be an increasing geometric progression of positive real numbers. If A_{1}A_{3}A_{5}A_{7}=\frac{1}{1296} and then, the value of A_{6}+A_{8}+A_{10}?
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Let A1, A2, A3,..... be an increasing geometric progression of positive real numbers. If A_{1}A_{3}A_{5}A_{7}=\frac{1}{1296} and then, the value of A_{6}+A_{8}+A_{10}? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let A1, A2, A3,..... be an increasing geometric progression of positive real numbers. If A_{1}A_{3}A_{5}A_{7}=\frac{1}{1296} and then, the value of A_{6}+A_{8}+A_{10}? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A1, A2, A3,..... be an increasing geometric progression of positive real numbers. If A_{1}A_{3}A_{5}A_{7}=\frac{1}{1296} and then, the value of A_{6}+A_{8}+A_{10}?.
Let A1, A2, A3,..... be an increasing geometric progression of positive real numbers. If A_{1}A_{3}A_{5}A_{7}=\frac{1}{1296} and then, the value of A_{6}+A_{8}+A_{10}? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let A1, A2, A3,..... be an increasing geometric progression of positive real numbers. If A_{1}A_{3}A_{5}A_{7}=\frac{1}{1296} and then, the value of A_{6}+A_{8}+A_{10}? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A1, A2, A3,..... be an increasing geometric progression of positive real numbers. If A_{1}A_{3}A_{5}A_{7}=\frac{1}{1296} and then, the value of A_{6}+A_{8}+A_{10}?.
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