Question

Calculati.a)27-25+23-21+....+7-5+3-1;b)28-26+24-22+....+8-6+4-2

Answer 1

Răspuns:

Explicație pas cu pas:

27-25+23-21+...+7-5+3-1= 3+7+11+15+19+23+27-1-5-9-13-17-21-25=

10+30+50+15=105-91=14

-10-30-30-21=-91

2(-1+2-3+4-...-13+14)= 14 termeni, fiecare 2 grupati =1

2 x 7= 14

Answer 2

$\displaystyle a)~27-25+23-21+...+7-5+3-1=\\ \\ =(3+7+11+...+27)-(1+5+9+...+25)\\ \\ \bullet\bullet\bullet\\ \\ \triangleright3+7+11+...+27 \\ \\ a_n=a_1+(n-1)\cdot r~~~~~~~~~~~~~~~~~~~~~~a_n=27;~a_1=3;~r=4\\\\27=3+(n-1)\cdot4\Rightarrow 27=3+4n-4\Rightarrow4n=27-3+4\Rightarrow 4n=28\Rightarrow\\\\ \Rightarrow n=\frac{28}{4}\Rightarrow n=7\\\\S_n=\frac{a_1+a_n}{2}\cdot n\\\\ S_{27}=\frac{3+27}{2} \cdot7=\frac{30}{2}\cdot 7=15\cdot7=105$

$\displaystyle 3+7+11+...+27=105\\ \\ \bullet\bullet\bullet\\ \\ \triangleright1+5+9+...+25\\ \\ a_n=a_1+(n-1)\cdot r~~~~~~~~~~~~~~~~~~~~~~~~~~a_n=25;~a_1=1;~r=4\\ \\ ~25=1+(n-1)\cdot4\Rightarrow25=1+4n-4\Rightarrow4n=25-1+4\Rightarrow4n=28\Rightarrow \\ \\ \Rightarrow n=\frac{28}{4}\Rightarrow n=7\\ \\ S_n=\frac{a_1+a_n}{2}\cdot n\\ \\ S_{25}=\frac{1+25}{2} \cdot 7=\frac{26}{2} \cdot7=13\cdot7=91$

$\displaystyle 1+5+9+...+25=91\\ \\ \bullet\bullet\bullet\\\\27-25+23-21+...+7-5+3-1=\\ \\=(3+7+11+...+27)-(1+5+9+...+25)=105-91=\mathbf{14}$

$\displaystyle b)~28-26+24-22+...+8-6+4-2=\\ \\ =(4+8+12+...+28)-(2+6+10+...+26)\\ \\ \bullet\bullet\bullet\\ \\ \triangleright4+8+12+...+28\\ \\ a_n=a_1+(n-1)\cdot r~~~~~~~~~~~~~~~~~~~~~~~~~~a_n=28;~a_1=4;~r=4~\\ \\ 28=4+(n-1)\cdot4\Rightarrow 28=4+4n-4\Rightarrow 4n=28-4+4\Rightarrow4n=28\Rightarrow \\ \\ \Rightarrow n=\frac{28}{4}\Rightarrow n=7\\ \\ S_n=\frac{a_1+a_n}{2} \cdot n \\ \\ S_{28}=\frac{4+28}{2} \cdot7=\frac{32}{2} \cdot7=16\cdot7=112$

$\displaystyle 4+8+12+...+28=112\\ \\ \bullet\bullet\bullet\\ \\ \triangleright 2+6+10+...+26 \\ \\ a_n=a_1+(n-1)\cdot r ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a_n=26;~a_1=2;~r=4~\\ \\ 26=2+(n-1)\cdot 4\Rightarrow 26=2+4n-4\Rightarrow 4n=26-2+4\Rightarrow 4n=28\Rightarrow \\ \\ \Rightarrow n=\frac{28}{4} \Rightarrow n=7\\ \\ S_n=\frac{a_1+a_n}{2}\cdot n\\ \\ S_{26}=\frac{2+26}{2} \cdot7=\frac{28}{2}\cdot 7=14\cdot7=98$

$\displaystyle 2+6+10+...+26=98\\ \\ \bullet\bullet\bullet\\ \\28-26+24-22+...+8-6+4-2=\\ \\ =(4+8+12+...+28)-(2+6+10+...+26)=112-98=\mathbf{14}$