# Between 1 and 31, m arithmetic means have been inserted in such a way that the ratio of the 7th  and (m - 1)th means is 5:9. Find the value of m.

Let the means be x1, x2 , x3 , x4 ,……, xm.

So that 1, x1, x2, x3, x4,……, xm, 31 is an A.P of m + 2 terms

am + 2 = a + (m + 1)d

31 = 1 + (m + 1)d

(m + 1)d = 30

d = 30/(m + 1) ………………….(i)

x7 : xm - 1 = 5:9

a + 7d : a + (m - 1)d = 5 : 9

9(1 + 7d) = 5(1 + (m - 1)d)

9 + 63d = 5 + 5md - 5d

4 = -63d + 5md - 5d

4 = (-68 + 5m)d

4 = (-68 + 5m)[30/ (m +1)]

4(m + 1) = (-68 + 5m)30

2m + 2= -1020 + 75m

73m = 1022

m = 14

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