Question

с |+2|+|+51 +|+81 +...+|+29|=
d -2¹+|-2|2+...+|-2|20=

Answer 1

Răspuns:

Explicație pas cu pas:

c)  fiind termeni pozitivi, putem elimina modulul si adaugam cate o unitate fiecarui termen, ca sa ajungem la o suma Gauss

3+6+9+......+30 -10=3(1+2+3+...+10)-10=3·10·11/2-10=3·5·11-10=

=15·11-10=165-10=155

d)

modulul va face o suma de puteri ale lui 2

2+2²+2³+.....+2^20

S=2·(2^20-1)/(2-1)=2(2^20-1)=2(2^10+1)(2^5+1)(2^5-1)=

 =2·(2^10+1)(32+1)(32-1)=2·33·31(2^10+1)=2046×1025=2 097 150

Answer 2

Explicație pas cu pas:

c)

$| + 2| + | + 5| + | + 8| + |11| + ... + | + 29| = \\ = 2 + 5 + 8 + 11 + ... + 29 \\ = 2 + (2 + 1 \cdot 3) + (2 + 2 \cdot 3) + (2 + 3 \cdot 3) + ... + (2 + 9 \cdot 3) \\ = \underbrace{2 + 2 + 2 + 2 + ... + 2}_{10} + 3 \cdot (1 + 2 + 3 + ... + 9) \\ = 2 \cdot 10 + 3 \cdot \frac{9 \cdot 10}{2} = 20 + 135 = \bf 155$

d)

$S = { | - 2| }^{1} + { | - 2| }^{2} + { | - 2| }^{3} + ... + { | - 2| }^{20} \\ = {2}^{1} + {2}^{2} + {2}^{3} + ... + {2}^{20}$

$S = {2}^{1} + {2}^{2} + {2}^{3} + ... + {2}^{20}$

$2S = 2 \cdot ( {2}^{1} + {2}^{2} + {2}^{3} + ... + {2}^{20})$

$2S + 2 = \underbrace{{2}^{1} + {2}^{2} + {2}^{3} + {2}^{4} + ... + {2}^{20}}_{S} + {2}^{21}$

$2S - S + 2 = {2}^{21} \implies \bf S = {2}^{21} - 2$