1+3+5+7+9+...+99=
= 1+2+3+...+99 - ( 2+4+6+...+98)=
[tex] = \frac{99 \times 100}{2} - 2 \times (1 +2 + ... + 49) = [/tex]
[tex] = 99 \times 50 - 2 \times \frac{49 \times 50}{2} = [/tex]
[tex] = 99 \times 50 - 49 \times 50 = [/tex]
[tex]50\times (99 - 49) = [/tex]
[tex] = 50 \times 50 = {50}^{2} [/tex]
