Question

IV va rog muuuuuult!!!

Answer 1

a = 1 + 1 / 2¹ + 1/ 2² + 1 / 2³ + 1 / 2⁴

cmmmc al numitorilor 2¹;  2²;   2³   si 2⁴ = 2⁴ = 16

a = ( 2⁴ + 2³ + 2² + 2¹ + 1 ) / 2⁴

a = ( 16 + 8 + 4 + 2 + 1 ) / 16

a = 31 / 16

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b = 1 + 1 / 2¹ + 1 / 2² + 1 / 2³ + 1 / 2⁴ + 1 / 2⁵

cmmmc al numitorilor este 2⁵ = 32

b = ( 2⁵ + 2⁴ + 2³ + 2² + 2¹ + 1 ) / 2⁵

b = ( 32 + 16 + 8 + 4 + 2 + 1 ) / 32

b = 63 / 32

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a )   a = 2 - 1 / 2ˣ

2  - 1 / 2ˣ = 31 / 16

2ˣ = 16 = 2⁴ = >   x = 4

Verific:

2 - 1 / 2⁴ = ( 2⁵ - 1 ) / 16 = ( 32 - 1 ) / 16 = 31 / 16 = a

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b)   b - a = 63 / 32 - 31 / 16 = 63 / 32 - 62 / 32 = 1 / 32

b + 2 / 2⁵ = 2 + 1 / 2⁵

63 / 32 + 2 / 32 = 65 / 32

2 + 1 / 2⁵ = ( 2 x 2⁵ + 1 ) / 2⁵ = ( 2⁶ + 1 ) / 2⁵ = ( 64 + 1 ) / 32 = 65 / 32

==>   b + 2 / 2⁵ = 2 + 1 / 2⁵ = 65 / 32

b = 63 / 32 = 1,96875  < 2 ==>  b < 2