Question

Calculati1+3+5+...+105

Answer 1

1+3+5+...+105=53²=2809
Formula:(2n-1)=n²
2n-1=105
2n=105+1
2n=106
n=106:2
n=53

Answer 2

$\displaystyle 1+3+5+...+105= \\ \\ =1+2+3+4+5+...+105-(2+4+6+...+104)= \\ \\ = \frac{105(105+1)}{2} -2(1+2+3+...+52)= \\ \\ = \frac{105 \times 106}{2} -2 \times \frac{52(52+1)}{2} = \frac{11130}{2} -2 \times \frac{52 \times 53}{2} = \\ \\ =5565- \not 2 \times \frac{2756}{ \not 2} =5565-2756=2809$