Which is larger number , `99^(100) + 100^(50)` or `101^(50)` ?

Options

A. `99^(50) + 100^(50)`
B. both are equal
C. ` (101)^(50)`
D. none of these

Answer

c

Explanation

We have ,
` 99^(50) + 100 ^(50) - 101^(50)`
` = (100 - 1 )^(50) - (1 - 100)^(50) + 1000^(50)`
` = - {(1 +100)^(50) - (1 - 100)^(50)} + 100^(50)`
` = - 2 {""^(50)C_(1)xx100 +""^(50)C_(3)xx100^(3) + ... + ""^(50)C_(49)xx100^(49)} + 100^(50)`
` = - 2 {""^(50)C_(1)xx100 +""^(50)C_(3)xx100^(3) + ... + ""^(50)C_(47)xx100^(49)} lt 0`
` therefore 99^(50) + 100^(50) - 101 ^(50) lt 0 rArr 00^(50) + 100^(50) lt 101^(50)`.