Question

Calculati urmatoarele sume:
a) S= 1+3+5+7+...+97+99
b) S= 2+4+6+8+...+100+102

Answer 1

$\displaystyle a).1+3+5+7+...+97+99 \\ \\ 99=1+(n-1) \times 2 \\ \\ 99=1+2n-2 \\ \\ 2n=99-1+2 \\ \\ 2n=100 \\ \\ n= \frac{100}{2} \\ \\ n=50$

$\displaystyle S_{50}= \frac{2+49 \times 2}{2} \times 50 \\ \\ S_{50}= \frac{2+98}{2} \times 50 \\ \\ S_{50}= \frac{100}{2} \times 50 \\ \\ S_{50}=50 \times 50 \\ \\ S_{50}=2500$

$\displaystyle b).2+4+6+8+...+100+102=2(1+2+3+...+50+51)= \\ \\ =2 \times \frac{51(51+1)}{2} =2 \times \frac{51 \times 52}{2} =\not 2\times \frac{2652}{\not 2} =2652$