Question

(an) n>1 a3=11 ; a11=3 . Calculați a7=?

Answer 1

$\displaystyle \displaystyle a_3=11,~ a_{11}=3,~a_7=? \\ a_3=11 \Rightarrow a_{3-1}+r=11 \Rightarrow a_2+r=11 \Rightarrow a_1+2r=11 \Rightarrow \\ \Rightarrow a_1=11-2r \\ a_{11}=3 \Rightarrow a_{11-1}+r=3 \Rightarrow a_{10}+r=3 \Rightarrow a_1+10r=3 \Rightarrow \\ \Rightarrow 11-2r+10r=3 \Rightarrow -2r+10r=3-11 \Rightarrow 8r=-8 \Rightarrow r=-1 \\ a_1=11-2r \Rightarrow a_1=11-2 \cdot (-1) \Rightarrow a_1=11+2 \Rightarrow a_1=13$
$a_7=a_{7-1}+r \Rightarrow a_7=a_6+r \Rightarrow a_7=a_1+6r \Rightarrow a_7=13+6 \cdot (-1) \Rightarrow \\ \Rightarrow a_7=13-6 \Rightarrow \boxed{a_7=7}$