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Question: The larger of \(99^{50} + 100^{50}\) and \(101^{50}\) is...

The larger of 9950+1005099^{50} + 100^{50} and 10150101^{50} is

A

9950+1005099^{50} + 100^{50}

B

Both are equal

C

10150101^{50}

D

None of these

Answer

10150101^{50}

Explanation

Solution

We have,

10150=(100+1)50=10050+50.10049+50.492.110048+101^{50} = (100 + 1)^{50} = 100^{50} + 50.100^{49} + \frac{50.49}{2.1}100^{48} +.....(i)

And 9950=(100−1)50=10050−50.10049+50.492.110048−.......99^{50} = (100 - 1)^{50} = 100^{50} - 50.100^{49} + \frac{50.49}{2.1}100^{48} - ...........(ii)

Subtracting, 10150−9950=10050+2.50.49.483.2.110047+.....>10050101^{50} - 99^{50} = 100^{50} + 2.\frac{50.49.48}{3.2.1}100^{47} + ..... > 100^{50}.

Hence 10150>10050+9950101^{50} > 100^{50} + 99^{50}.