Question

1+3+5+7+... +19 =?
1+3+5+7+....+99 = ?

Answer 1

$\displaystyle a).1+3+5+7+...+19 \\ 19=1+(n-1) \times 2 \\ 19=1+2n-2 \\ 2n=19-1+2 \\ 2n=20 \\ n=20:2 \\ n=10 \\ S_{10}= \frac{2+9 \times 2}{2} \times 10 \\ \\ S_{10}= \frac{2+18}{2} \times 10 \\ \\ S_{10}= \frac{20}{2} \times 10 \\ \\ S_{10}=10 \times 10 \\ S_{10}=100$

$\displaystyle b).1+3+5+7+...+99 \\ 99=1+(n-1) \times 2 \\ 99=1+2n-2 \\ 2n=99-1+2 \\ 2n=100 \\ n=100:2} \\ n=50$
$\displaystyle S_{50}= \frac{2+49 \times 2}{2} \times 50 \\ \\ S_{50}= \frac{2+98}{2} \times 50 \\ \\ S_{50}= \frac{100}{2} \times 50 \\ \\ S_{50}=50 \times 50 \\ S_{50}=2500$