Number of rational terms in the expansion of -2-4100 is

General term in the expansion of nbsp;(√(2)+√(4))^100 will be nbsp; T_r+1=^100C_r(2)^(100 r)/2(4)^r/5 ⇒ T_r+1=^100C_r(2)^(500 r)/10 Now for the rational ter ...

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Standard XIII

Mathematics

Greatest Binomial Coefficients

Number of rat...

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Number of rational terms in the expansion of ${(\sqrt{2} + \sqrt[5]{4})}^{100}$ is

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Solution

General term in the expansion of ${(\sqrt{2} + \sqrt[5]{4})}^{100}$ will be
$T_{r + 1} =^{100} C_{r} (2)^{(100 - r) / 2} (4)^{r / 5} \Rightarrow T_{r + 1} =^{100} C_{r} (2)^{\frac{(500 - r)}{10}}$
Now for the rational term $\frac{500 - r}{10}$ has to be an integer
$∴ r = 0, 10, 20, \dots, 100$
Hence there will be $11$ rational terms

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Number of rational terms in the expansion of (√2+5√4)100 is

General term in the expansion of (√2+5√4)100 will be Tr+1=100Cr(2)(100−r)/2(4)r/5⇒Tr+1=100Cr(2)(500−r)10 Now for the rational term 500−r10 has to be an integer ∴r=0,10,20,⋯,100 Hence there will be 11 rational terms

Number of rational terms in the expansion of ${(\sqrt{2} + \sqrt[5]{4})}^{100}$ is