# Find n if m=3 and the arithmetic mean of f(m) and f(n) is 15,5 for f(x)=x/3 + 10.

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We need to find n if m=3 and the arithmetic mean of f(m) and f(n) is 15,5 for f(x)=x/3 + 10.

Now if m = 3, f(m) = f(3) = 3/3 + 10 = 11.

Therefore the value of f(x) point corresponding to x=3 is 11.

Now we are given the mean of f(m) and (f(n) as 15.5.

Now f(m) = 11

So, 15.5*2 = f(m) + f(n) = 11 + f(n)

=> f(n) = 31 - 11 = 20

Now f(n) = n/3 + 10 = 20

=> n/3 = 20 - 10

=> n/3 = 10

=> n = 30

Therefore n = 30.

**The required value of n is 30. **

We see that the arithmetic mean of f(30) and f(3) is 15.5

We'll determine the arithmetic mean of the numbers f(m) and f(n):

A.M. = [f(m) + f(n)]/2

We'll substitute m and n in the expression of f(x).

For m = 3, we'll get:

f(3) = 3/3 + 10

f(3) = 1 + 10

**f(3) = 11 = f(m)**

For n, we'll get:

f(n) = n/3 + 10

Now, we'l substitute f(m) and f(n) by the values of f(3) and the expression of f(n), in the expression of A.M.

A.M. = [f(3) + f(n)]/2

A.M. = (11 + n/3 + 10)/2

But the A.M. = 15.5

15.5 = (21 + n/3)/2

We'll cross multiply:

31 = 21 + n/3

We'll multiply all terms by 3:

93 = 63 + n

We'll subtract 63 both sides and we'll use the symmetric property:

n = 93 - 63

**n = 30**

**For n = 30 and m = 3, the arithmetic mean of f(30) and f(3) is 15.5.**

f(x) = x/3+10.

Therefore f(m) = m/3. and f(n) = n/3+10.

Therefore for m=3, f(m) = f(5) = 5/3+10 = 35/3.

Therefore the arith matic mean of f(m) and f(n) = (f(m+f(n)}/2.

{f(m)+f9n)}/2 = {35/3 + n/3+10}/2 is given to be 15.5.

Therefore (35/3+n/3+10)/2 = 15.5.

Mulltiply both sides by 6:

35 +n +30 = 15.5*6 = 93

n = 93-65 = 28.

Therefore the value of n is 28.