Let the coefficients of the middle terms in the expansion of

Question

Let the coefficients of the middle terms in the expansion of (frac1sqrt(6)+beta x)^4,(1-3 beta x)^2 and (1-(beta / 2) x)^6, beta>0, respectively form the first three terms of an A.P. If d is the common difference of this A.P. , then 50-frac2 dbeta^2 is equal to __________.

DivineJEE · Accepted Answer

Coefficients of middle terms of given expansions are

{}^4{C_2}{1 \over 6}{\beta ^2},\,{}^2{C_1}( - 3\beta ),\,{}^6{C_3}{\left( {{{ - \beta } \over 2}} \right)^3}

form an A.P.

\therefore 2.2( - 3\beta ) = {\beta ^2} - {{5{\beta ^3}} \over 2}

\Rightarrow - 24 = 2\beta - 5{\beta ^2}

\Rightarrow 5{\beta ^2} - 2\beta - 24 = 0

\Rightarrow 5{\beta ^2} - 12\beta + 10\beta - 24 = 0

\Rightarrow \beta (5\beta - 12) + 2(5\beta - 12) = 0

\beta = {{12} \over 5}

d = - 6\beta - {\beta ^2}

\therefore 50 - {{2d} \over {{\beta ^2}}} = 50 - 2{{( - 6\beta - {\beta ^2})} \over {{\beta ^2}}} \\ = 50 + {{12} \over \beta } + 2 = 57

Explanation

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