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  • In the expansion of <img alt="\small \left(5^{1 / 2}+7^{1 / 8}\right)^{1024}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Csmall%20%5Cleft%285%5E%7B1%20/%202%7D&plus;7%5E%7B1%20/%208

In the expansion of \small \left(5^{1 / 2}+7^{1 / 8}\right)^{1024}, the number of integral terms is
 

Option: 1

128


Option: 2

129


Option: 3

130


Option: 4

131


Answers (1)

best_answer

    Here  n=1024=2^{10}, a power of 2, where as the power of 7 is  \frac{1}{8}=2^{-3}
        Now first term { }^{1024} C_0\left(5^{1 / 2}\right)^{1024}=5^{512} (integer)
    And after 8 terms, the 9th term   ={ }^{1024} C_8\left(5^{1 / 2}\right)^{1016}\left(7^{1 / 8}\right)^8an integer
        Again, 17th term = { }^{1024} C_{16}\left(5^{1 / 2}\right)^{1008}\left(7^{1 / 8}\right)^{16}
                         = An integer. 
        Continuing like this, we get an A.P. 1, 9, 17, ...., 1025,
        because 1025th term = the last term in the expansion
  ={ }^{1024} C_{1024}\left(7^{1 / 8}\right)^{1024}=7^{128}    (an integer)
        If n is the number of terms of above A.P. we have 1025=T_n=1+(n-1) 8 \Rightarrow n=129
        .
 

Posted by

Irshad Anwar

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