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

If p arithmetic means are inserted between a and b, thenb= (p+2)th term of AP series such thata= first term andd=common differenceNow (p+2)th term=a+(p+2-1)db=a+(p+1)db-a=(p+1)dd=b-ap+1

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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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