Arithmetic Mean

To insert a given number of arithmetic mean between two give quantities:-
let a and b be the two given quantities , n the number of means. Including the extremes the number of terms will be (n+2) ; so that we have find a series of(n+2) terms in A.P. of which a is the first term and b is the last or (n+2) term.
Let d be the common difference
b= the (n+2)th term = a + (n+1)d
b-a = (n+1)d;
(b-a)/(n+1) = d;
so the required means are a+(b-a)/(n+1) , a+2(b-a)/(n+1).....................

Q: Find the five arithmetic means between 5 and 25.
Ans: Here given that first term a=5 last term b = 25 and number of arithmetic mean n=5
Let d be the common difference of the required A.P.
so , b= a + (n+1)d;
putting the values of a ,b and n in the equation,we get
25= 5 + (5+1)d;
20= 6d;
d= 10/3;
Now the arithmetic means are 5+10/3,5+2*10/3,5+30/3,5+40/3 and 5+ 50/3
i.e. 25/3,35/3,45/3,55/3 and 65/3 are arithmetic means between 5 and 25.
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