The sum, of the coefficients of the first 50 terms in the binomial expansion of (1-x)^100,…

Question

The sum, of the coefficients of the first 50 terms in the binomial expansion of $(1-x)^{100}$, is equal to

Accepted Answer

$$
\begin{aligned}
& \left({ }^{100} C_0-{ }^{100} C_1+{ }^{100} C_2-\ldots . .{ }^{100} C_{49}\right)+{ }^{100} C_{50} \\
& +\left(-{ }^{100} C_{51}+{ }^{100} C_{52}-\ldots .+{ }^{100} C_{100}\right)=0 \\
& \lambda_1+{ }^{100} C_{50}+\lambda_2=0 \\
& \lambda_1=-\frac{1}{2}{ }^{100} C_{50} \quad\left(\because \lambda_1=\lambda_2\right) \\
& =-{ }^{99} C_{49}
\end{aligned}
$$

Solutions · 1