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  • Let  and [x] denotes the greatest integer less than or equal to x. If the sum of (n+1) terms

Let n \in N and [x] denotes the greatest integer less than or equal to x. If the sum of (n+1) terms { }^n C_0, 3 \cdot{ }^n C_1, 5 \cdot{ }^n C_2, 7{ }^n C_3, \ldots is equal to 2^{150} \cdot 151, \text{then}\ 2\left[\frac{n-1}{2}\right] is equal to

Option: 1

150
 


Option: 2

148
 


Option: 3

149
 


Option: 4

147


Answers (1)

best_answer

We have,

\begin{aligned} & \quad{ }^n C_0+3 \cdot{ }^n C_1+5 \cdot{ }^n C_2+\ldots \ldots+(2 n+1) \cdot{ }^n C_n=2^{150} \cdot 151 \\ & \therefore \quad T_r=(2 r+1)^n C_r \\ & \text { Let } S=\sum_{r=0}^n T_r=\sum_{r=0}^n(2 r+1)^n C_r=\sum_{r=0}^n 2 r{ }^n C_r+\sum_{r=0}^n{ }^n C_r \\ & \Rightarrow \quad S=2\left(n \cdot 2^{n-1}\right)+2^n=2^n(n+1) \\ & \text { Thus, } 2^n(n+1)=2^{150} \cdot 151 \Rightarrow n=150 \\ & \therefore \quad 2\left[\frac{n-1}{2}\right]=2\left[\frac{149}{2}\right]=148 . \end{aligned}

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