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Question

Mathematics Question on Binomial theorem

If x=9950+10050x = 99^{50} + 100^{50} and y=(101)50y = \left(101\right)^{50} then

A

x=yx = y

B

x<yx < y

C

x>yx > y

D

None of these

Answer

x<yx < y

Explanation

Solution

(101)50−(99)50=(100+1)50−(100−1)50\left(101\right)^{50}- \left(99\right)^{50} = \left(100 + 1\right)^{50}- \left(100-1\right) ^{50} =2[50C1(100)49+50C3(100)47+......+50C49(100)]= 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} ⇒(101)50>(99)50+(100)50⇒y>x⇒x<y.\Rightarrow \left(101\right)^{50} > \left(99\right)^{50} + \left(100\right)^{50} \Rightarrow y > x \Rightarrow x < y .