Question:

If $x = 99^{50} + 100^{50}$ and $y = \left(101\right)^{50}$ then

Updated On: Jul 6, 2022
  • $x = y$
  • $x < y$
  • $x > y$
  • None of these
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The Correct Option is B

Solution and Explanation

$\left(101\right)^{50}- \left(99\right)^{50} = \left(100 + 1\right)^{50}- \left(100-1\right) ^{50}$ $= 2\left[^{50}C_{1}\left(100\right)^{49}+^{50}C_{3}\left(100\right)^{47}+...... + ^{50}C_{49} \left(100\right)\right]$ $> 2. ^{50}C_{1} . \left(100\right)^{49} = 2 \times50\left(100\right)^{49} = \left(100\right)^{50}$ $\Rightarrow \left(101\right)^{50} > \left(99\right)^{50} + \left(100\right)^{50} \Rightarrow y > x \Rightarrow x < y .$
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Notes on Binomial theorem

Concepts Used:

Binomial Theorem

The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is 

Properties of Binomial Theorem

  • The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
  • There are (n+1) terms in the expansion of (x+y)n.
  • The first and the last terms are xn and yn respectively.
  • From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
  • The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.