Let [t] denote the greatest integer <= t. If the constant term in the expansion of (3 x^2-frac12 x^5)^7 is alpha, then [alpha] is equal to ___________.

Let T_r+1 be the constant term. T_r+1= ^7 C_r(3 x^2)^7-r((-1 / 2 x^5))^r For constant term, power of x should be zero. i.e., 14-2 r-5 r=0 \ => 14=7 r => r=2 Now, constant term =alpha => ^7 C_2(3)^5((-1 / 2))^2=alpha \ => 21 x 243 x (1 / 4)=alpha \ => [alpha]=[1275.75]=1275

Let t denote the greatest integer leq t If the constant... | null (Mathematics)

Let

\mathrm{T}_{r+1}

be the constant term.

\mathrm{T}_{r+1}={ }^7 \mathrm{C}_r\left(3 x^2\right)^{7-r}\left(\frac{-1}{2 x^5}\right)^r

For constant term, power of

x

should be zero.

\begin{aligned} & \text { i.e., } 14-2 r-5 r=0 \\\\ & \Rightarrow 14=7 r \Rightarrow r=2 \end{aligned}

Now, constant term

=\alpha

\begin{aligned} & \Rightarrow{ }^7 C_2(3)^5\left(\frac{-1}{2}\right)^2=\alpha \\\\ & \Rightarrow 21 \times 243 \times \frac{1}{4}=\alpha \\\\ & \Rightarrow[\alpha]=[1275.75]=1275 \end{aligned}

Explanation

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