Question

Pls ajutati-ma ex 13 14 17.ofer 30 puncte va rog frumos

Answer 1

13)

2(1 + 3 + 3^2 + ...+ 3^50 < 3^51

a = 3^0 + 3^1 + 3^2 + ...+ 3^49 + 3^50 | × 3

3 × a = 3^1 + 3^2 + 3^3 +...+ 3^50 + 3^51

3×a - a = 3^51 - 3^0

2 × a = 3^51 - 1

a = (3^51 - 1) : 2

2 × (3^51 - 1) : 2 = 3^51 - 1 < 3^51 (adevarat)
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14)
a = 1 + 2 + 2^2 + 2^3 +...+ 2^100 + 2^101

a = 2^0 + 2^1 + 2^2 + 2^3 +...+ 2^100+2^101 |×2

2×a = 2^1+2^2+2^3+2^4+...+2^101+2^102

2×a - a = 2^102 - 2^0

a = 2^102 - 1
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b = 3(1 + 4 + 4^2 + ... + 4^50)

notam paranteza cu n

n = 4^0 + 4^1 + 4^2 + ... + 4^49 + 4^50 |×4

4×n = 4^1 + 4^2 + 4^3 + ... + 4^50 + 4^51

4×n - n = 4^51 - 4^0

3 × n = 4^51 - 1

n = (4^51 - 1) : 3

b = 3 ( 4^51 - 1) : 3

b = 4^51 - 1

b = (2^2)^51 - 1

b = 2^102 - 1
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=> 2^102 - 1 = 2^102 - 1 => a = b
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17)
a) (4 + 3)^3 - (4 - 3)^10 = 7^3 - 1^10 = 343-1 = 342

b) (10^3-125×2^3) : 123^123 =

(1000 - 125 × 8) : 123^123 =

(1000 - 1000) : 123^123 =

0 : 123^123 =

0
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c) [(10^3:5^2:8+3^2×5^2):3^0]×2^2^0 =

[(1000:25:8+9×25): 1] × 2^1 =

(5 + 225) × 2 =

230 × 2 =

460
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d) {[(12^2+5^2) : 13]^2 - 2^7 - 2^0} : 2^3 =

{[(144 + 25) : 13]^2 - 128 - 1} : 8 =

[( 169 : 13)^2 - 128 - 1] : 8 =

( 13^2 - 128 - 1) : 8 =

(169 - 128 - 1) : 8 =

40 : 8 =

5
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