n harmonic means have been inserted between 1 and 4 such that first meant:last mean=1:3,find n.
this question answer is 11 sir that answer prove
geez - you ever going to show some of your own work? These slow-mo homework dumps are annoying.
To solve this problem, we first need to understand what is meant by the "first mean" and "last mean" being in a particular ratio.
The n harmonic means inserted between 1 and 4 form a set of n+1 numbers. The harmonic mean of two numbers is defined as the reciprocal of the arithmetic mean of their reciprocals. So, if we have numbers x and y, the harmonic mean (H) between them is given by:
H = 2/(1/x + 1/y)
Now, let's break down the given information: "first mean:last mean = 1:3". This means that the first mean is one-third the value of the last mean.
Let's denote the first mean as x, and the last mean as y. According to the given information:
x/y = 1/3
Now, we need to find a specific value for x and y using the harmonic mean formula. We know that the numbers 1 and 4 are the extreme terms, and we need to insert n harmonic means between them.
Let's calculate the arithmetic mean (A) of the extremes:
A = (1 + 4)/2 = 5/2 = 2.5
Now, we can use the harmonic mean formula:
1st harmonic mean = reciprocal of A = 1/2.5 = 2/5
2nd harmonic mean = reciprocal of x = 1/x
3rd harmonic mean = reciprocal of y = 1/y
...
Using the given ratio x/y = 1/3, we can express x and y in terms of the reciprocal of the harmonic means as:
x = (1/3)(1/y) = 1/(3y)
y = 3x
Since the number of harmonic means is n, we have (n+1) terms in total (including the two extremes). Therefore, we can express the sum of the terms as:
Sum = 1 + (1/2.5) + 1/x + (1/y) + ...
To find n, we need to equate this sum to 4 (the sum of the extreme terms):
1 + (1/2.5) + 1/x + (1/y) + ... = 4
Now, substitute the expressions for x and y:
1 + (1/2.5) + 1/(1/(3y)) + (1/y) + ... = 4
Simplifying this equation will give us the value of n.
Note: This equation involves the reciprocal of the harmonic means, but by taking their reciprocals, we are essentially finding the harmonic means themselves.
I apologize, but the calculation for n in this specific problem involves solving an equation with a series of reciprocals, which can be quite complex to do manually. You may need to use numerical methods or a calculator to find the value of n.