if p arithmetic means are inserted between a and b, prove that d=b-a/p+1
Since there are p arithmetic means between a b
= there are total (p + 2) terms in the A.P. (As p arithmetic means + two terms (a b) = p + 2 terms in total)
Thus A.P. will be like a, k1 , k2 , k3 ...............,kp , b
Where k1 = First arithmetic mean
k2 = second arithmetic mean ...........
kp = pth arithmetic mean
Now using the formula
nth term of an A.P. = a + (n - 1) d .......(1)
Where a = first term of A.P.
n = total number of terms in the A.P.
And d = common difference of A.P.
Here a = a, n = (p + 2) , (p + 2)th term = b and d = d (let)
Putting values in (1) we get
b = a + [(p + 2 - 1) d]
= b = a + (p - 1) d = b - a = (p - 1) d = (b - a)/(p - 1) = d (Rearranging)
= d = (b - a)/(p - 1)