${{a}_{1}}=1=\frac{1\times 2}{2},{{a}_{2}}=2+1=3=\frac{2\times 3}{2},{{a}_{3}}=3+2+1=6=\frac{3\times 4}{2}\Rightarrow {{a}_{n}}=\frac{n(n+1)}{2}\Rightarrow {{a}_{4}}=\frac{4(4+1)}{2}=10$
$10+\frac{n(n+1)}{2}=38\Rightarrow \frac{n(n+1)}{2}=28\Rightarrow n(n+1)=56\Rightarrow {{n}^{2}}+n=56$
${{n}^{2}}+n-56=0\Rightarrow \Delta ={{b}^{2}}-4ac=1-4\times 1\times (-56)=225$
$_{{{n}_{1}}=\frac{-b-\sqrt{\Delta }}{2a}=\frac{-1-15}{2}=-8}^{{{n}_{1}}=\frac{-b+\sqrt{\Delta }}{2a}=\frac{-1+15}{2}=7}$