There are n arithmetic means between 1 and 31 such that seventh mean:(n-1)th mean =5:9. Find n.

If there are n means, then the common difference is (31-1)/(n+1) = 30/(n+1)

The kth mean is 1+k*30/n

So, you want

[1+7*30/(n+1)]/[1+(n-1)*30/(n+1)] = 5/9
n = 14

Check:
1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31

To solve this problem, let's first understand what is meant by "n arithmetic means" between 1 and 31.

In mathematics, an arithmetic mean refers to the average of a set of numbers. In this case, we need to find the arithmetic means between 1 and 31. So, these arithmetic means will form a sequence of numbers with equal differences between them.

Let's assume that the first mean is x and the common difference between the means is y.

The arithmetic sequence will then be:
1, x, (x + y), (x + 2y), ... , (x + (n-1)y), 31

Now, based on the given information, we know that the ratio between the seventh mean and the (n-1)th mean is 5:9. Mathematically, this can be expressed as:

(x + 6y) / (x + (n-2)y) = 5/9

To find the value of n, we need to solve this equation.

Multiplying both sides of the equation by 9(x + (n-2)y), we get:

9(x + 6y) = 5(x + (n-2)y)

Expanding both sides of the equation, we have:

9x + 54y = 5x + 5ny - 10y

Combining like terms, we get:

4x + 64y = 5ny

Simplifying further, we have:

4x = 5ny - 64y

Factoring out y on the right-hand side, we get:

4x = y(5n - 64)

Now, there can be multiple values of x and y that satisfy this equation. However, we are interested in finding the value of n, which represents the number of arithmetic means.

To find n, we need to determine the smallest integer value that satisfies the equation. This means that we need to find the smallest possible value of y that satisfies the equation for some integer values of x.

Let's consider the expression (5n - 64) for various values of n.

For n = 1, the expression (5n - 64) equals -59.
For n = 2, the expression (5n - 64) equals -54.
For n = 3, the expression (5n - 64) equals -49.
For n = 4, the expression (5n - 64) equals -44.
For n = 5, the expression (5n - 64) equals -39.
For n = 6, the expression (5n - 64) equals -34.

Beyond this point, if we increase the value of n, the expression (5n - 64) will become positive. Therefore, the smallest possible positive value of y (as y should be a positive value to represent the common difference between the means) will occur when n = 7.

Substituting n = 7 into the equation, we have:

4x = y(5(7) - 64)
4x = y(35 - 64)
4x = y(-29)

Since y needs to be a positive value, the only possible solution is when y = 1, which gives:

4x = 1(-29)
4x = -29
x = -29/4

However, since we are looking for arithmetic means between 1 and 31, x cannot be negative. Hence, there are no valid solutions for x and y when n = 7.

Therefore, there are no positive integer values of n that satisfy the given condition.