Q. There are n GM's between 1/25 and 25. if the ratio of first mean to third mean is 1:25.find n.

Soln:

here,

no. of GM's (n) = ?

first term (a) =1/25

last term (b) =25

first mean/third mean = 1/25

we have,

common ratio (r) =(b/a)^(1/n+1)

r = {25/(1/25)}^(1/n+1)

r = (625)^(1/n+1)

r = (5)^(4/n+1)..............equation(i)

also we have,

GM1/GM3 = 1/25

ar/ar^3 = 1/25

1/r^2 = 1/25

r^2 = 25

r^2 = 5^2

r = 5............equation(ii)

put the value of r = 5 from equation(ii) into equation(i), we get

5 = (5)^(4/n+1)

1 = 4/(n+1)

n + 1 = 4

n = 4 - 1

n = 3 Ans.