Question

cine mă ajuta să-mi facă o parte din exerciti și va rog sa numerotați exercitiul​

Answer 1

$\displaystyle 2)~~1+6+11+...+x=1404\\\\ S_n=\frac{n[2a_1+(n-1)\cdot r]}{2} \\\\ a_1=1,~r=6-1=7,~S_n=1404\\\\ 1404=\frac{n[2 \cdot 1+(n-1) \cdot 5]}{2} \Rightarrow 1404=\frac{n(2+5n-5)}{2} \Rightarrow \\\\ \Rightarrow 1404 \cdot 2=2n+5n^2-5n \Rightarrow 2808 =2n+5n^2-5n \Rightarrow \\\\\Rightarrow -5n^2-2n+5n+2808=0\Rightarrow -5n^2+3n+2808=0\Big|\cdot(-1)\Rightarrow \\\\\Rightarrow 5n^2-3n-2808=0\\\\ \Delta=(-3)^2-4 \cdot 5 \cdot (-2808)=9+56160=56169 > 0$

$\displaystyle n_1=\frac{-(-3)-\sqrt{56169} }{2 \cdot 5} =\frac{3-237}{10} =-\frac{234}{10} =-\frac{117}{5} \not \in~\mathbb{N}$

$\displaystyle n_2=\frac{-(-3)+\sqrt{56169} }{2 \cdot 5} =\frac{3+237}{10} =-\frac{240}{10} =24\in\mathbb{N}$

$\displaystyle 3)~~1+8+15+...+x=1956\\\\ S_n=\frac{n[2a_1+(n-1)\cdot r]}{2} \\\\ a_1=1,~r=8-1=7,~S_n=1956\\\\ 1956=\frac{n[2 \cdot 1+(n-1) \cdot 7]}{2} \Rightarrow 1956=\frac{n(2+7n-7)}{2} \Rightarrow \\\\\Rightarrow 1956\cdot 2=2n+7n^2-7n\Rightarrow 3912=2n+7n^2-7n\Rightarrow \\\\\Rightarrow -7n^2-2n+7n+3912=0\Rightarrow -7n^2+5n+3912=0\Big|\cdot(-1)\Rightarrow \\\\\Rightarrow 7n^2-5n-3912=0\\\\\Delta=(-5)^2-4 \cdot 7 \cdot (-3912)=25+109536=109561$

$\displaystyle n_1=\frac{-(-5)-\sqrt{109561} }{2 \cdot 7} =\frac{5-331}{14} =-\frac{326}{14} =-\frac{163}{7} \not\in\mathbb{N}\\\\n_2=\frac{-(-5)+\sqrt{109561} }{2 \cdot 7} =\frac{5+331}{14} =\frac{336}{14} =24 \in\mathbb{N}$

$4)~~a_4+a_{15}=135,~a_8+a_{11}=?\\\\ a_4=a_1+(4-1)\cdot r=a_1+3r\\\\ a_{15}=a_1+(15-1) \cdot r=a_1+14r\\\\ a_4+a_{15}=135 \Rightarrow a_1+3r+a_1+14r=135\Rightarrow 2a_1+17r=135\\\\ a_8+a_{11}=a_1+(8-1) \cdot r+a_1+(11-1) \cdot r=a_1+7r+a_1+10r=\\\\=2a_1+17r=135$

$5)~a_6+a_{23}=153,~a_{10}+a_{19}=?\\\\ a_6+a_{23}=153\Rightarrow a_1+(6-1)\cdot r+a_1+(23-1)\cdot r=153\Rightarrow\\\\\Rightarrow a_1+5r+a_1+22r=153\Rightarrow 2a_1+27r=153\\\\ a_{10}+a_{19}=a_1+(10-1)\cdot r+a_1+(19-1)\cdot r=a_1+9r+a_1+18r=\\\\=2a_1+27r=153$

$6)~a_3+a_{22}=194,~a_{11}+a_{14}=?\\\\ a_3+a_{22}=194\Rightarrow a_1+(3-1)\cdot r+a_1+(22-1)\cdot r=194 \Rightarrow \\\\\Rightarrow a_1+2r+a_1+21r=194\Rightarrow 2a_1+23r=194\\\\a_{11}+a_{14}=a_1+(11-1)\cdot r+a_1+(14-1)\cdot r=a_1+10r+a_1+13r=\\\\=2a_1+23r=194$

$7)~~a_1,~a_2,~30,~42,...\\\\ r=42-30=12\\\\ a_3=a_2+r \Rightarrow a_2=a_3-r \Rightarrow a_2=30-12\Rightarrow a_2=18\\\\ a_2=a_1+r\Rightarrow a_1=a_2-r\Rightarrow a_1=18-12\Rightarrow a_1=6$

$8)~~a_1,~a_2,~34,~46,...\\\\r=46-34=12\\\\ a_3=a_2+r \Rightarrow a_2=a_3-r \Rightarrow a_2=34-12\Rightarrow a_2=22\\\\ a_2=a_1+r \Rightarrow a_1=a_2-r\Rightarrow a_1=22-12 \Rightarrow a_1=10$

$9)~~a_1,~a_2,~33,~44,...\\\\r=44-33=11\\\\ a_3=a_2+r \Rightarrow a_2=a_3-r \Rightarrow a_2=33-11\Rightarrow a_2=22\\\\ a_2=a_1+r \Rightarrow a_1=a_2-r\Rightarrow a_1=22-11 \Rightarrow a_1=11$

$10)~~a_7+a_{10}=132,~r=8,~a_1=?\\\\ a_7+a_{10}=132 \Rightarrow \displaystyle a_1+6r+a_1+9r=132 \Rightarrow 2a_1+15r=132 \Rightarrow \\\\ \Rightarrow 2a_1+15 \cdot 8=132 \Rightarrow 2a_1+120=132 \Rightarrow 2a_1=132-120 \Rightarrow 2a_1=12 \Rightarrow \\\\ \Rightarrow a_1=\frac{12}{2} \Rightarrow a_1=6$

$\displaystyle 11)~~a_{13}+a_{19}=68,~r=2,~a_1=?\\\\ a_{13}+a_{19}=68 \Rightarrow a_1+12r+a_1+18r=68 \Rightarrow 2a_1+30r=68 \Rightarrow\\\\ \Rightarrow 2a_1+30 \cdot 2=68 \Rightarrow 2a_1+60=68\Rightarrow 2a_1=68-60 \Rightarrow 2a_1=8 \Rightarrow \\\\\Rightarrow a_1=\frac{8}{2} \Rightarrow a_1=4$

$\displaystyle 12)~~a_7+a_{24}=213,~r=7,~a_1=?\\\\ a_{7}+a_{24}=213 \Rightarrow a_1+6r+a_1+23r=213\Rightarrow 2a_1+29r=213\Rightarrow \\\\ \Rightarrow 2a_1+29 \cdot 7=213\Rightarrow 2a_1+203=213\Rightarrow 2a_1=213-203 \Rightarrow 2a_1=10\Rightarrow \\\\\Rightarrow a_1=\frac{10}{2} \Rightarrow a_1=5$

$\displaystyle13)~~a_2=7,~a_{26}=79\\\\\left\{\begin{array}{ccc}a_2=7\\a_{26}=79\end{array}\right \Rightarrow\left\{\begin{array}{ccc}a_1+r=7\Big|\cdot(-1)\\a_1+25r=79\end{array}\right\Rightarrow \left\{\begin{array}{ccc}-a_1-r=-7\\a_1+25r=79\end{array}\right \\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~---------\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~/~~~24r=72 \Rightarrow r=\frac{72}{24} \Rightarrow r=3\\\\a_1+r=7\Rightarrow a_1+3=7 \Rightarrow a_1=7-3 \Rightarrow a_1=4$