Question

13 Calculați:
a. |-1| + |-2| + |-3| +...+ |-50|
b. |-3| + |-6| + |-9| +...+ |-90|
(c si d in poza:)​

Answer 1

Răspuns:

a) 1275; b) 1395; c) 155; d) 2²¹ - 2

Explicație pas cu pas:

a)

$| - 1| + | - 2| + | - 3| + ... + | - 50| = 1 + 2 + 3 + ... + 50 =$

$= \frac{50 \cdot 51}{2} = \bf 1275$

b)

$| -3| + | - 6| + | - 9| +...+ | - 90| = 3 + 6 + 9 + ... + 90 =$

$= 3 \cdot (1 + 2 + 3 + ... + 30) = 3 \cdot \frac{30 \cdot 31}{2} = \bf 1395$

c)

$| + 2| + | + 5| + | + 8| +...+ | + 29| = 2 + 5 + 8 + ... + 29 = (2 + 29) + (5 + 26) + (8 + 23) + (11 + 20) + (14 + 17) = 5 \cdot 31 = \bf 155$

d)

${ | - 2| }^{1} + { | - 2| }^{2} + { | - 2| }^{3} + ... + { | - 2| }^{20} = {2}^{1} + {2}^{2} + {2}^{3} + ... + {2}^{20} = \bf {2}^{21} - 2$

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

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

$2S = {2}^{2} + {2}^{3} + {2}^{4} + ... + {2}^{20} + {2}^{21}$

$2S + 2 = 2 + {2}^{2} + {2}^{3} + {2}^{4} + ... + {2}^{20} + {2}^{21}$

$2S + 2 = ({2}^{1} + {2}^{2} + {2}^{3} + {2}^{4} + ... + {2}^{20}) + {2}^{21}$

$2S + 2 = S + {2}^{21}$

$\red {\bf S = {2}^{21} - 2}$