Question

4. Find the arithmetic mean of :
(i) -5 and 41


(iii) (m + n)2 and (m - n)2​

Answer 1

Let N>1 be a positive integer. What will be the Arithmetic mean of positive integers less than N and co-prime with N?

Getting no idea how to proceed!

(The way the problem reads at this time is "What will be the Arithmetic mean of positive integers less than N and co-prime with N?" (I mention this in case there was some other version of the question and it got edited.))

n is coprime to N if and only if N−n is coprime to N.

The average of n and N−n is N/2.

The average of the averages of all such cases is the average of a bunch of numbers each equal to N/2.

The average of all numbers in {1,…,N} that are coprime to N is therefore N/2.

Later edit: Here's an example. The numbers in {1,…,20} that are coprime to 20 are 1,3,7,9,11,13,17,19.

They come in these pairs:

1,3,7,9,19 (The average of these two is 10.)17 (The average of these two is 10.)13 (The average of these two is 10.)11 (The average of these two is 10.)

Now take the average of all those 10s. The average is 10.

Answer 2

Step-by-step explanation:

(i) arthematic mean of -5,41

= -5+41/2

= 36/2

= 18

(ii) (m+n)2 , (m-n)2

= 2m+2n+2m-2n/2

= 2m+2m+2n-2n/2

= 4m/2

= 2m