Question

lf p arithmetic means are inserted between a and b, prove that d =
b-a/p+1
​

Answer 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)

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