1.

9. Show that the product of n geometric means between a and b is equal to the nth power of the single GM between a and b.

Answer» Let `G_(1), G_(2), ... G_(n)` be n GMs between a and b, and let r be its common ratio. Then, `a, G_(1), G_(2), ..., G_(n), b` are in GP.
`:. T_(n+2)=b rArr ar^((n+2-1))=b rArr r^((n+1))=(b/a) rArr r=(b/a)^(1/((n+1)))` ...(i)
`:. G_(1)xxG_(2)xxG_(3)xx...xxG_(n)=(ar)xx(ar^(2))xx...xx(ar^(n))`
`=a^(n)xxr^((1+2+...+n))=a^(n)xxr^(1/2n (n+1))`
`=a^(n)xx(b/a)^(n//2)=a^(n//2)xxb^(n//2)` [using (i)]
`=(sqrt(ab))^(n)=G^(n)`, where `G=sqrt(ab)`.


Discussion

No Comment Found

Related InterviewSolutions