Question

calculati urmatoarele sume. 3+7+11+15+...+43

Answer 1

Observam ca primul termen se mai poate scrie sub forma 3*1+0, al doilea termen 3*2+1..(voi nota "a indice 1" cu a1 etc)

a1=3*1+0

a2=3*2+1

a3=3*3+2

........

a43=3*43+42

Adunam pe coloane si vom obtine suma lui Gauss:

3*(1+2+...43)+(1+2+...+42)=3*43*44/2+42*43/2=2838+903=3741

Answer 2

$\displaystyle 3+7+11+15+...+43 \\ \\ 43=3+(n-1) \times 4 \\ \\ 43=3+4n-4 \\ \\ 4n=43-3+4 \\ \\ 4n=44 \\ \\ n= \frac{44}{4} \\ \\ n=11$

$\displaystyle S_{11}= \frac{6+10 \times 4}{2} \times 11 \\ \\ S_{11}= \frac{6+40}{2} \times 11 \\ \\ S_{11}= \frac{46}{2} \times 11 \\ \\ S_{11}=23 \times 11 \\ \\ S_{11}=253$