Question

Calculati
$\sqrt{4x + 1}$
Stiind ca
$x = 1 + 5 + {5}^{2} + {5}^{3} + ... + {5}^{2012} + {5}^{2013}$



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Answer 1

Răspuns:

Explicație pas cu pas:

x=1+5+5²+5³+...+5²⁰¹²+5²⁰¹³

5x= 5+5²+5³+...+5²⁰¹²+5²⁰¹³+5²⁰¹⁴, scadem parte cu parte,

5x-x=5+5²+5³+...+5²⁰¹²+5²⁰¹³+5²⁰¹⁴-(1+5+5²+5³+...+5²⁰¹²+5²⁰¹³)=5²⁰¹⁴-1

Deci 4x=5²⁰¹⁴-1, atunci 4x+1=5²⁰¹⁴-1+1=5²⁰¹⁴

$\sqrt{4x+1} =\sqrt{5^{2014}}=\sqrt{(5^{1007})^{2}} =5^{1007}$