A motor has 6 speeds varying from 16rpm to 256 rpm in a gp calculate the common ratio

In a geometric progression (GP), the common ratio \( r \) is the factor by which each term is multiplied to get the next term. The speeds range from 16 rpm to 256 rpm, with 6 speeds in total.

In a GP, the \( n \)-th term is given by:

\[
T_n = T_1 \times r^{n-1}
\]

Where:
- \( T_n \) is the \( n \)-th term,
- \( T_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

Here, \( T_1 = 16 \) rpm, \( T_6 = 256 \) rpm, and \( n = 6 \). Substituting into the equation for the \( 6 \)-th term:

\[
256 = 16 \times r^{6-1}
\]

\[
256 = 16 \times r^5
\]

Dividing both sides by 16:

\[
16 = r^5
\]

Now, solve for \( r \) by taking the 5th root of 16:

\[
r = \sqrt[5]{16}
\]

Let me calculate this for you.

\[
r = 2
\]

Thus, the common ratio \( r \) is 2.

Would you like more details on how this is derived, or have any questions?

Here are some related questions to explore:
1. What is a geometric progression (GP)?
2. How does the common ratio affect a geometric sequence?
3. What happens if the common ratio is less than 1?
4. How can you determine the sum of terms in a geometric progression?
5. Can geometric progressions model other real-world scenarios?

**Tip:** In geometric sequences, the common ratio can be found by dividing any term by the previous one.

A motor has 6 speeds varying from 16rpm to 256rpm in a geometric progression. Calculate the common ratio.

Learn how to calculate the common ratio in a geometric progression for a motor with 6 speeds ranging from 16 rpm to 256 rpm. This step-by-step solution covers geometric sequences, exponents, and algebra, suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation