Упр.13.32 ГДЗ Мерзляк Поляков 7 класс (Алгебра)

1) a^(n+1)+a^n;
2) b^n-b^(n-3),n > 3;
3) c^(n+2)+c^(n-4),n > 4;
4) d^2n-d^n;
5) 2^(n+3)+3•2^(n+2)-5•2^(n+1);
6) 9^(n+1)+3^(n+2);
7) 49^(n+2)-7^(n+3);
8) 64^(n+2)-32^(n+2)+16^(n+2).

1. $$a^{n+1}+a^n=a^n(a+1)$$
2. $$b^n-b^{n-3}=b^{n-3}(b^3-1)$$
3. $$c^{n+2}+c^{n-4}=c^{n-4}(c^6+1)$$
4. $$d^{2n}-d^n=d^n(d^n-1)$$
5. $$2^{n+3}+3\cdot 2^{n+2}-5\cdot 2^{n+1}=2^{n+1}(2^2+3\cdot 2-5)=2^{n+1}(4+6-5)=5\cdot 2^{n+1}$$
6. $$9^{n+1}+3^{n+2}=(3^2)^{n+1}+3^{n+2}=3^{2n+2}+3^{n+2}=3^{n+2}(3^n+1)$$
7. $$49^{n+2}-7^{n+3}=(7^2)^{n+2}-7^{n+3}=7^{2n+4}-7^{n+3}=7^{n+3}(7^{n+1}-1)$$
8. $$64^{n+2}-32^{n+2}+16^{n+2}=(2^6)^{n+2}-(2^5)^{n+2}+(2^4)^{n+2}$$
   $$ =2^{6n+12}-2^{5n+10}+2^{4n+8} =2^{4n+8}\left(2^{2n+4}-2^{n+2}+1\right) $$

Ответ

1) $$a^n(a+1)$$; 2) $$b^{n-3}(b^3-1)$$; 3) $$c^{n-4}(c^6+1)$$; 4) $$d^n(d^n-1)$$; 5) $$5\cdot 2^{n+1}$$; 6) $$3^{n+2}(3^n+1)$$; 7) $$7^{n+3}(7^{n+1}-1)$$; 8) $$2^{4n+8}\left(2^{2n+4}-2^{n+2}+1\right)$$.