Find the largest

Algebra Level 3

The larger of $A=99^{ 50 }+100 ^{ 50 }$ and $B=101^{ 50 }$ is

Both are equal B A Cannot be determined

1 solution

Suyash Mittal
May 8, 2014

$Consider,\quad { (101) }^{ 50 }-{ (99) }^{ 50 }-{ (100) }^{ 50 }$ $={ (100+1) }^{ 50 }-{ (100-1) }^{ 50 }-{ (100) }^{ 50 }$ $=\{ { (100) }^{ 50 }{ (1+0.01) }^{ 50 }-{ (1-0.01) }^{ 50 }-1\}$ $={ (100) }^{ 50 }\{ 2\{ _{ }^{ 50 }{ { C }_{ 1 } }(0.01)+_{ }^{ 50 }{ { C }_{ 3 }{ (0.01) }^{ 3 }+...\} -1\} }$ $={ (100) }^{ 50 }\{ 2\{ _{ }^{ 50 }{ { C }_{ 3 } }{ (0.01) }^{ 3 }+_{ }^{ 50 }{ { C }_{ 5 }{ (0.01) }^{ 5 }+...\} }$ $\therefore \quad { (101) }^{ 50 }-\{ { (99) }^{ 50 }+{ (100) }^{ 50 }\} \quad >\quad 0$ $\Rightarrow \quad { (101) }^{ 50 }\quad >\quad { (99) }^{ 50 }+{ (100) }^{ 50 }$

Find the largest

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