всего вариантов 8/4 вариантов когда 4 е подряд=5
<span> (138+517)</span>²<span> >138</span>²<span>+517</span>², так как (138+517)²= 138²+517²+2·138·517
![y_n= \frac{3n+10}{3-4n} \\\\y_1= \frac{3*1+10}{3-4*1}= \frac{3+10}{3-4}= \frac{13}{-1}=-13 \\\\y_2= \frac{3*2+10}{3-4*2}= \frac{6+10}{3-8}= \frac{16}{-5}=-3,2 \\\\y_3= \frac{3*3+10}{3-4*3}= \frac{9+10}{3-12}= \frac{19}{-9}=-2 \frac{1}{9} \\\\y_4= \frac{3*4+10}{3-4*4}= \frac{12+10}{3-16}= \frac{22}{-13}=-1 \frac{9}{13} \\\\\\\\ y_n\ \textgreater \ -1\\\\\frac{3n+10}{3-4n}\ \textgreater \ -1\\\\\frac{3n+10}{3-4n}+1\ \textgreater \ 0\\\\\frac{3n+10+3-4n}{3-4n}\ \textgreater \ 0\\\\ \frac{13-n}{4(0,75-n)}\ \textgreater \ 0](https://tex.z-dn.net/?f=y_n%3D+%5Cfrac%7B3n%2B10%7D%7B3-4n%7D+%5C%5C%5C%5Cy_1%3D+%5Cfrac%7B3%2A1%2B10%7D%7B3-4%2A1%7D%3D+%5Cfrac%7B3%2B10%7D%7B3-4%7D%3D+%5Cfrac%7B13%7D%7B-1%7D%3D-13++%5C%5C%5C%5Cy_2%3D+%5Cfrac%7B3%2A2%2B10%7D%7B3-4%2A2%7D%3D+%5Cfrac%7B6%2B10%7D%7B3-8%7D%3D+%5Cfrac%7B16%7D%7B-5%7D%3D-3%2C2++%5C%5C%5C%5Cy_3%3D+%5Cfrac%7B3%2A3%2B10%7D%7B3-4%2A3%7D%3D+%5Cfrac%7B9%2B10%7D%7B3-12%7D%3D+%5Cfrac%7B19%7D%7B-9%7D%3D-2+%5Cfrac%7B1%7D%7B9%7D+++%5C%5C%5C%5Cy_4%3D+%5Cfrac%7B3%2A4%2B10%7D%7B3-4%2A4%7D%3D+%5Cfrac%7B12%2B10%7D%7B3-16%7D%3D+%5Cfrac%7B22%7D%7B-13%7D%3D-1+%5Cfrac%7B9%7D%7B13%7D+%5C%5C%5C%5C%5C%5C%5C%5C+y_n%5C+%5Ctextgreater+%5C+-1%5C%5C%5C%5C%5Cfrac%7B3n%2B10%7D%7B3-4n%7D%5C+%5Ctextgreater+%5C+-1%5C%5C%5C%5C%5Cfrac%7B3n%2B10%7D%7B3-4n%7D%2B1%5C+%5Ctextgreater+%5C+0%5C%5C%5C%5C%5Cfrac%7B3n%2B10%2B3-4n%7D%7B3-4n%7D%5C+%5Ctextgreater+%5C+0%5C%5C%5C%5C+%5Cfrac%7B13-n%7D%7B4%280%2C75-n%29%7D%5C+%5Ctextgreater+%5C+0++++)
+ - +
///////////////////////////(0,75)_______________(13)///////////////////////////
Учитывая, что n∈N (натуральное число), получаем, что n>13
Следовательно, начиная с n=14 все члены последовательности будут больше, чем -1.