1.

There are n arithmetic means between 9 and 27 . If the ratio of the last mean to the first mean is 2:1, find the value of n.

Answer» Correct Answer - n=5
Let `A_(1),A_(2),….,A_(n)` be the arithmetic between 9 and 27.
Then , 9 , ` A_(1),A_(2),…..,A_(n) ` , 27 are in AP.
9+( n+2-1) d = 27` Rightarrow d = 18/(n+1)`
` A_(n)/A_(1)=2/1 Rightarrow T_(n-1)/T_(2) = 2/1 Rightarrow ( 9+(n+1-1)d)/(9 +(2-1) d) =2/1`
` Rightarrow 9+ nd=18+2d Rightarrow d= 9/(n-2)`
` 18/(n+2) = 9/(n-2) Rightarrow 18(n-2) = 9(n+1) Rightarrow n=5`


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