QJEEM15Maff32: JEE Main 2015 Mathematics - Class 11 JEEM15Maff32 Solution: If m is the A.M. of two distinct real numbers l and n and and…

Question

Verified Solution • 2025–26 Session moderate

If m is the A.M. of two distinct real numbers l and n

(l,n &gt; 1)

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

equals:

Accepted Answer

m = {{l + n} \over 2}

and common ratio of

G.P.

= r = {\left( {{n \over l}} \right)^{{1 \over 4}}}

\therefore

{G_1} = {l^{3/4}}\,{n^{1/4}},

{G_2} = {l^{1/2}}{n^{1/2}},\,

\,\,\,\,\,\,\,\,\,\,\,\,\,\,{G_3} = {l^{1/4}}{n^{3/4}}

G_1^4 + 2G_2^4 + G_3^4

= {l^3}n + 2{l^2}{n^2} + {\ln ^3}

= \ln {\left( {1 + n} \right)^2}

= \ln \times 2{m^2}

= 4l{m^2}n

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