Question

Fie a=1+6+6^2+...+6^20 si b=1+5+5^2+....+5^22. Comparati 5a cu 4b

Answer 1

a = 6^0 + 6^1 + 6^2 + ... + 6^19 + 6^20 | × 6

6 × a = 6^1 + 6^2 + 6^3 +...+ 6^20 + 6^21

6×a - a = 6^21 - 6^0

5×a = 6^21 - 1

a = ( 6^21 - 1)/5
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b = 5^0 + 5^1 + 5^2 +...+ 5^21 + 5^22 |× 5

5 × b = 5^1 + 5^2 + 5^3 +...+ 5^22 + 5^23

5×b - b = 5^23 - 5^0

4 × b = 5^23 - 1

b = (5^23 - - 1) / 4
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5a = 5 × (6^21 - 1) / 5 = 6^21 - 1

4b = 4 × (5^23 - 1) / 4 = 5^23 - 1

6^21 - 1 = (6^3)^7 - 1 = 216^7 - 1

5^23 - 1 = 5^21 × 5^2 - 1 =
(5^3)^7 × 25 - 1 = 125^7 × 25 - 1

216 > 125 =>

=> 216^7 - 1 > 125^7 × 25 - 1

=> 6^21 - 1 > 5^23 - 1
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