Question

$\sqrt{1 + 3 + 5 + 7 + ...... + 101}$

Answer 1

√(1+3+5+7+...+101) =
√( 50×51+51) =
√(51×(50+1)) =
√ (51×51) =
51

Answer 2

$\sqrt{1+3+5+7.....+101}= \\ \\ 1+3+5+................. +101=$

3=2+1
5=4+1
7=6+1
.....
101=100+1
1+3+..........+101=1+2+1+4+1+6+1+......+100+1=51×1+2(1+2+.............+50)
=51+2×[(50×51)/2]=51+50×51=51(1+50)=51²

$1+2+3+................. +101=51^{2}$

$\sqrt{1+3+5+.......+101}= \sqrt{51^{2} }=51$