Упр.1098 ГДЗ Алимов 10-11 класс (Алгебра)

Упростить:

1. $$\frac{(n+3)!}{(n+1)!}=\frac{(n+3)(n+2)(n+1)!}{(n+1)!}=(n+3)(n+2).$$

2. $$\frac{(n+2)!}{(n-1)!}=\frac{(n+2)(n+1)n(n-1)!}{(n-1)!}=n(n+1)(n+2).$$

3. $$\left(\frac{1}{(n+1)!}+\frac{1}{n!}\right)\cdot n!=\frac{n!}{(n+1)!}+\frac{n!}{n!}=\frac{1}{n+1}+1=\frac{n+2}{n+1}.$$

4. $$\left(\frac{1}{n!}-\frac{1}{(n+1)!}\right)\cdot n!=\frac{n!}{n!}-\frac{n!}{(n+1)!}=1-\frac{1}{n+1}=\frac{n}{n+1}.$$

5. $$\left(\frac{1}{n!}-\frac{1}{(n+2)!}\right)\cdot (n+1)!=\frac{(n+1)!}{n!}-\frac{(n+1)!}{(n+2)!}=(n+1)-\frac{1}{n+2}.$$

   $$=(n+1)-\frac{1}{n+2}=\frac{(n+1)(n+2)-1}{n+2}=\frac{n^2+3n+1}{n+2}.$$

6. $$\left(\frac{1}{(n+2)!}+\frac{1}{n!}\right)\cdot (n+1)!=\frac{(n+1)!}{(n+2)!}+\frac{(n+1)!}{n!}=\frac{1}{n+2}+(n+1).$$

   $$\frac{1}{n+2}+(n+1)=\frac{1+(n+1)(n+2)}{n+2}=\frac{n^2+3n+3}{n+2}.$$