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  • Solve! - Binomial theorem and its simple applications - JEE Main

The sum of coefficients of integral powers of  x  in  the  binomial  expansion  of

\left ( 1-2\sqrt{x} \right )^{50}\; is:

  • Option 1)

    \frac{1}{2}\left ( 3^{50}+1 \right )

  • Option 2)

    \frac{1}{2}\left ( 3^{50} \right )

  • Option 3)

    \frac{1}{2}\left ( 3^{50}-1 \right )

  • Option 4)

    \frac{1}{2}\left ( 2^{50}+1 \right )

 

Answers (1)

As we learnt in

Properties of Binomial Theorem -

\left ( x+a \right )^{n}+\left ( x-a \right )^{n}= 2\left ( ^{n}c_{0} \, x^{n}+ ^{n}c_{2}\, x^{n-2}\, a^{2}+---\right )

- wherein

Sum of odd terms or even Binomial coefficients

 

 \left(1-2\sqrt{x} \right )^{50}=1-^{50}C_{1}\ 2\sqrt{x}+^{50}C_{2}\ (2\sqrt{x})^{2}-..................

\underline\left(1+2\sqrt{x} \right )^{50}=1-^{50}C_{1}\ 2\sqrt{x}+^{50}C_{2}\ (2\sqrt{x})^{2}+..................

Add and put x = 1

1+3^{50}=2 \left(1+^{50}C_{2}\ 2x+............. \right )

Sum of integral powers = =\frac{3^{50}+1}{2}

Correct option is 1.

 


Option 1)

\frac{1}{2}\left ( 3^{50}+1 \right )

This is the correct option.

Option 2)

\frac{1}{2}\left ( 3^{50} \right )

This is an incorrect option.

Option 3)

\frac{1}{2}\left ( 3^{50}-1 \right )

This is an incorrect option.

Option 4)

\frac{1}{2}\left ( 2^{50}+1 \right )

This is an incorrect option.

Posted by

Sabhrant Ambastha

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