if the mean of first n natural numbers 5n/9 then n is

The mean of the first \( n \) natural numbers is given by the formula:

\[
\text{Mean} = \frac{\text{Sum of the first } n \text{ natural numbers}}{n}
\]

The sum of the first \( n \) natural numbers is:

\[
S_n = \frac{n(n+1)}{2}
\]

So the mean is:

\[
\text{Mean} = \frac{\frac{n(n+1)}{2}}{n} = \frac{n+1}{2}
\]

According to the problem, this mean is also given as \( \frac{5n}{9} \). Therefore, we can set up the equation:

\[
\frac{n+1}{2} = \frac{5n}{9}
\]

Now solve for \( n \).

1. Multiply both sides by 18 to eliminate the denominators:

\[
9(n+1) = 10n
\]

2. Expand and simplify:

\[
9n + 9 = 10n
\]

\[
9 = 10n - 9n
\]

\[
n = 9
\]

Thus, \( n = 9 \).

Would you like further details on the steps or have any questions?

Here are some related questions to deepen understanding:
1. How do you calculate the sum of the first \( n \) natural numbers?
2. What is the formula for the mean of a sequence of numbers?
3. Can you generalize the formula for the mean of the first \( n \) even numbers?
4. What would happen if the mean was \( \frac{6n}{10} \) instead of \( \frac{5n}{9} \)?
5. How can the arithmetic mean be used to solve other types of problems involving sequences?

**Tip:** In any arithmetic sequence, the mean is equal to the average of the first and last term!

if the mean of first n natural numbers is 5n/9 then n is

Learn how to solve the problem where the mean of the first n natural numbers is given as 5n/9. Using the sum formula for natural numbers and algebraic manipulation, we determine that n = 9. Suitable for students in 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