Question

se da progresia aritmetica a4=5 a7=11 sa se calculeze a14 și S (suma) 50. Am nevoie urgent

Answer 1

$\displaystyle a_4=5,~a_7=11,~a_{14}=?,~S_{50}=? \\ a_4=5 \Rightarrow a_{4-1}+r=5 \Rightarrow a_3+r=5 \Rightarrow a_1+3r=5 \Rightarrow a_1=5-3r \\ a_7=11 \Rightarrow a_{7-1}+r=11 \Rightarrow a_6+r=11 \Rightarrow a_1+6r=11 \Rightarrow \\ \Rightarrow 5-3r+6r=11 \Rightarrow 5+3r=11 \Rightarrow 3r=11-5 \Rightarrow 3r=6 \Rightarrow \\ \Rightarrow r= \frac{6}{3} \Rightarrow r=2 \\ a_1=5-3r \Rightarrow a_1=5-3 \cdot 2 \Rightarrow a_1=5-6 \Rightarrow a_1=-1$
$\displaystyle =a_{14}=a_{14-1}+r \Rightarrow a_{14} =a_{13}+r \Rightarrow a_{14}=a_1+13r \Rightarrow \\ \Rightarrow a_{14}=-1+13 \cdot 2 \Rightarrow a_{14}=-1+26 \Rightarrow a_{14}=25 \\ S_{50}= \frac{-2+49 \cdot 2}{2} \cdot 50 \Rightarrow S_{50}= \frac{-2+98}{2} \cdot 50\Rightarrow S_{50}= \frac{96}{2} \cdot 50 \Rightarrow \\ \\ S_{50}=48 \cdot 50 \Rightarrow S_{50}=2400$