Question

Va rog ex 1,2. Dau 98 de puncte si coroana, repede

Answer 1

a)   n = 6 x + 5 y

x = ( - 28 + 34 ) · [ ( - 7 ) ( + 6 ) - ( - 5 ) ( + 7 ) - ( - 18 ) : ( + 3 ) ]

x = 6 · ( - 42 + 35 + 6 )

x = 6 · ( - 7 + 6 )

x = 6 · ( - 1 )

x = - 6

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y = ( - 18 + 24 ) · [ ( - 4 ) ( - 16 ) + ( - 14 )· ( + 5 ) - ( - 32 ) : ( + 4 ) ]

y = 6 ·  (  64 - 70 + 8 )

y = 6 · ( - 6 + 8 )

y = 6 · 2

y = 12

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n = 6 x + 5 y = 6 · ( - 6 ) + 5 ·  12 = - 36 + 60 = 24

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b)

( - 96 ) : ( - 6 ) + [ ( - 66 + 26 ) : ( - 13 + 5 ) - ( - 75 + 11 ) : ( + 24 - 8 ) · ( - 2 ) = 13

= 16 + [ ( - 40 ) : ( - 8 ) - ( - 64 ) :  16 · ( - 2) =

= 16 + (  5 - 8 ) =

= 16 - 3 =

= 13

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c)

( - 3 ) · [ ( - 24 ) : ( + 3 ) + ( - 72 ) : ( + 9 ) - ( - 15 ) : ( + 3 ) = 33

= ( - 3 ) · [ ( - 8 ) - 8 + 5 ] =

= ( - 3 ) · ( - 16 + 5 ) =

= ( - 3 ) · ( - 11 ) =

= 33

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d)

[ ( - 32 ) : ( - 8 ) ( + 4 ) - ( - 27 ) : ( + 3 ) · ( - 2 ) ] : [ ( - 50 ) + ( - 8 ) ( - 6 ) ] = 1

= [ 4 · ( + 4 ) - ( - 9 ) · ( - 2 ) ] : [ - 50 ) + 48 ] =

= ( 16 - 18 ) : ( - 2 ) =

= ( - 2 ) : ( - 2 ) =

= 1

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2)

a)  5¹¹ ⁺ ¹² ⁺ ¹³ ⁺ ....... ⁺ ³⁰ = ( 5ⁿ ) ³ x 25

11 + 12 + 13 + ......+ 30 = 410

30 - 11 + 1 = 19 + 1 = 20 termeni are suma

= 20 x ( 11 + 30 ) / 2 =

=41 x 10 =  410

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( 5ⁿ ) ³ x 5² = 5⁴¹⁰

5 ³ⁿ ⁺ ² = 5 ⁴¹⁰

3 n + 2 = 41`0

3 n = 410 - 2 = 408

n = 408 : 3

n = 136

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b)

7²⁰ ⁺ ²⁴ ⁺ ²⁸ ⁺ ........ ⁺ ⁴⁸  = 49 ⁿ

20 + 24 + 28 + ......... + 48 = 272

( 48 - 20 ) : 4 + 1 = 28 : 4 + 1 = 7 + 1 = 8 termeni are suma cu ratia 4 ( 24 - 20;  28 - 24 = 4 )

= 8 x ( 20 + 48 ) / 2 =

= 68 x 4 =

= 272

49ⁿ = 7²⁷²

7²ⁿ = 7²⁷² == >  2 n = 272 ==>  n = 272 : 2 ==> n = 136

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c)

13¹ ⁺ ³ ⁺ ⁵ ⁺ ⁷ ⁺ ....... ⁺ ⁴⁹  = 13 ⁿ ²

1 + 3 + 5 + 7 + .......+ 49 =

( 49 - 1 ) : 2 + 1 = 48 : 2 + 1 = 24 + 1 = 25 termeni are suma

= 25 x ( 1 + 49 ) / 2 =

= 25 x 25 = 25²

13 ⁿ² = 13 ²⁵ ˣ ²⁵

n² = 25² ==>   n = √25² ==> n = 25

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d)

5ⁿ ⁺² + 5 ⁿ ⁺ ¹  - 15ⁿ : 3ⁿ = 29 x 25¹³

5ⁿ x 5² + 5ⁿ x 5 - ( 15 : 3 )ⁿ = 29 x 25¹³

5ⁿ ( 5² + 5 - 1 ) = 29 x 25¹³

5ⁿ x ( 25 + 5 - 1) = ( 5²) ¹³  x 29

5 ⁿ x 29 = 5 ⁽ ² ˣ ¹³ ⁾ x 29   l  : 29

5ⁿ = 5²⁶  == > n = 26

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e)

9ⁿ ⁺ ¹ x 9 ²ⁿ ⁺ ² = 27 ¹⁴

9 ⁽ ⁿ ⁺ ¹ ⁺ ²ⁿ ⁺ ² = ( 3³ )¹⁴

9 ³ⁿ ⁺ ³  = 3 ⁽ ³ ˣ ¹⁴ ⁾

( 3²) ³ ⁽ ⁿ ⁺ ¹ ⁾  = 3 ⁴²

3 ⁶ ⁽ ⁿ ⁺ ¹ ⁾ = 3 ⁴²  == >   n + 1 = 42 : 6 == > n = 7 - 1 == > n = 6