We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
+1
156
1
avatar

Let S be the sum of all the real coefficients of the expansion of \({(1+ix)}^{2009}\). What is \(\log_{2}(S)\)?

 Jan 25, 2019
 #1
avatar+6046 
+1

\(\text{The sum of all the real coefficients is simply the real part of }\\ \left . (1+ix)^{2009} \right |_{x=1}\)

 

\((1+i)^{2009} = \sqrt{2}^{2009}e^{i 2009 \left(\frac \pi 4\right)} = \sqrt{2}^{2009}\left(\cos\left(\dfrac \pi 4\right)+i \sin\left(\dfrac \pi 4 \right)\right)\)

 

\(Re\left[(1+i)^{2009}\right] = \sqrt{2}^{2009} \cdot \dfrac{\sqrt{2}}{2} = 2^{1004}\\ \log_2\left(2^{1004}\right) = 1004\)

.
 Jan 25, 2019
edited by Rom  Jan 25, 2019

18 Online Users

avatar
avatar
avatar