What is the geometric mean of 3, 9, 27, 81, 243, 729, 2187?
Understanding Geometric Mean
The geometric mean is a type of average that indicates the central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean, which uses their sum). It is calculated by multiplying all the numbers in a set and then taking the n-th root of the product, where n is the count of numbers in the set.
The formula for the geometric mean (GM) of a set of n numbers (\(x_1, x_2, \dots, x_n\)) is:
\( GM = \sqrt[n]{x_1 \times x_2 \times \dots \times x_n} \)
Applying Geometric Mean Calculation
The question asks for the geometric mean of the following sequence of numbers:
3, 9, 27, 81, 243, 729, 2187
First, let's count the numbers in the set. There are 7 numbers, so n = 7.
We can observe that these numbers are powers of 3:
- \(3 = 3^1\)
- \(9 = 3^2\)
- \(27 = 3^3\)
- \(81 = 3^4\)
- \(243 = 3^5\)
- \(729 = 3^6\)
- \(2187 = 3^7\)
Step-by-Step Calculation
- Identify the numbers and count (n): The numbers are \(3^1, 3^2, 3^3, 3^4, 3^5, 3^6, 3^7\). Here, \(n = 7\).
- Calculate the product of the numbers:
Product = \(3^1 \times 3^2 \times 3^3 \times 3^4 \times 3^5 \times 3^6 \times 3^7\)
Using the exponent rule \(a^m \times a^n = a^{m+n}\), we add the exponents:
Product = \(3^{(1+2+3+4+5+6+7)}\)
- Sum the exponents: The sum of the first 7 positive integers is calculated as \(\frac{n(n+1)}{2}\).
Sum of exponents = \(\frac{7(7+1)}{2} = \frac{7 \times 8}{2} = \frac{56}{2} = 28\).
So, the product is \(3^{28}\).
- Calculate the geometric mean: Now we take the 7th root of the product (\(3^{28}\)).
\( GM = \sqrt[7]{3^{28}} \)
Using the exponent rule \(\sqrt[n]{a^m} = a^{m/n}\):
\( GM = 3^{28/7} \)
\( GM = 3^4 \)
- Final Result: Calculate \(3^4\).
\(3^4 = 3 \times 3 \times 3 \times 3 = 81\).
Conclusion
The geometric mean of the numbers 3, 9, 27, 81, 243, 729, and 2187 is 81.
Similar Questions
- Let the positive numbers \(a_1, a_2, a_3, ...., a_{3n}\) be in GP. If P is the GM of \(a_1, a_2, a_3, ...., a_n\) and Q is the GM of \(a_{n+1}, a_{n+2}, a_{n+3}, ...., a_{3n}\), then what is the GM of \(3n\) numbers?
Important Questions from Geometric Progression
In a school of 750 boys, 34% are Muslims, 42% Hindus, 12% Sikhs and the rest belong to other communities. How many boys are from other community?
Ms. Tina has 2 parents, 4 grand parents, 8 great grand parents and so on. Find the number of her ancestors during 10 (ten) generations preceeding her own.
A square is drawn by joining the midpoints of the sides of a given square. In the same way, this process continues indefinitely. If side of the first square is 4 cm, than the sum of the area of all the squares are equal to
Third term of sequence an = (-1)n(n − 7) is :
811/3 × 811/9 × 811/27 × 811/81 × ______ up to infinite term will be equal to
