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If m is the AM of two distinct real numbers l and n `(l,ngt1)` and `G_(1),G_(2)" and "G_(3)` are three geometric means between l and n, then `G_(1)^(4)+2G_(2)^(4)+G_(3)^(4)` equalsA. `4l^(2)m^(2)n^(2)`B. `4l^(2)mn`C. `4lm^(2)n`D. `4lmn^(2)` |
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Answer» Correct Answer - C Given, `m=(l+n)/2rArr l+n =2n" ...(i)"` and l, `G_1, G_2, G_3,` n are in GP `:." "G_1/l =G_2/G_1=G_2/G_2=n/G_3` `rArr" "G_1G_3 = "In," G_1^2 = lG_2, G_2^2 =G_3 G_1, G_3^2 =nG_21" (ii)"` Now, `G_1^4 +G_2^4 +G_3^4 =l^2 G_2^2+2G_2^4+n^2G_2^2` `=G_2^2(l^2 +2G_2^2 +n^2)" "["from eq.(ii)"]` `=G_3 G_1 (l^2 +2G_2^2 +n^2) " " [from Eq.(ii) ]` `=G_3 G_1 (l^2 +2G_3 G_1 + n^2)` = In `(l^2+"2 In"+n^2)" " [from Eq.(ii)]` = In `(l+n)^2` = In `(2m)^2" "[from Eq. (i)]` `=4//m^2n` |
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