Nice Combination!

Which is greater: 9 9 50 + 10 0 50 99^{50}+100^{50} or 10 1 50 101^{50} ?

9 9 50 + 10 0 50 99^{50}+100^{50} 10 1 50 101^{50} Both are equal

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Chew-Seong Cheong
Mar 24, 2017

10 1 50 = ( 100 + 1 ) 50 = 10 0 50 + 50 10 0 49 + 50 49 2 10 0 48 + 50 49 48 2 3 10 0 47 + . . . 9 9 50 = ( 100 1 ) 50 = 10 0 50 50 10 0 49 + 50 49 2 10 0 48 50 49 48 2 3 10 0 47 + . . . 10 1 50 9 9 50 = 100 10 0 49 + 50 49 48 3 10 0 47 + . . . 10 1 50 = 9 9 50 + 10 0 50 + 50 49 48 3 10 0 47 + . . . 10 1 50 > 9 9 50 + 10 0 50 \begin{aligned} 101^{50} & = (100+1)^{50} = 100^{50} + 50\cdot 100^{49} + \frac {50\cdot49}2 \cdot 100^{48} + \frac {50\cdot49\cdot 48}{2\cdot 3} \cdot 100^{47} + ... \\ 99^{50} & = (100-1)^{50} = 100^{50} - 50\cdot 100^{49} + \frac {50\cdot49}2 \cdot 100^{48} - \frac {50\cdot49\cdot 48}{2\cdot 3} \cdot 100^{47} + ... \\ 101^{50} - 99^{50} & = {\color{#3D99F6}100 \cdot 100^{49}} + \frac {50\cdot49\cdot 48}3 \cdot 100^{47} + ... \\ 101^{50} & = 99^{50} + {\color{#3D99F6}100^{50}} + \frac {50\cdot49\cdot 48}3 \cdot 100^{47} + ... \\ \implies 101^{50} & \boxed{>} \ 99^{50} + 100^{50} \end{aligned}

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...