Question

Va rog ajutați-mă am mare nevoie, cu multe puncte și coroana!

Answer 1

Răspuns: n = 54

Explicație pas cu pas:

Salutare!

$\bf 2^{1} + 2^{2} +2^{3} + 2^{4}+....+2^{n} = 2^{55} - 2$

$\bf S = 2^{1} + 2^{2} +2^{3} + 2^{4}+....+2^{n} \:\:\:\Big|\cdot 2$

$\bf 2 \cdot S = 2^{1}\cdot 2 + 2^{2}\cdot 2 +2^{3}\cdot 2 + 2^{4}\cdot 2+....+2^{n}\cdot 2$

$\bf 2S = 2^{2} + 2^{3} +2^{4} + 2^{5}+....+2^{n+1}$

$\bf 2S = (2^{1} +2^{2} + 2^{3} +2^{4}+....+2^{n}) +2^{n+1} - 2^{1}$

$\bf 2S = S +2^{n+1} - 2^{1}$

$\bf 2S = S +2^{n+1} - 2$

$\bf 2S -S = 2^{n+1} - 2$

$\bf S = 2^{n+1} - 2$

$\bf 2^{55} = 2^{n+1} \implies 55 = n+1 \implies n = 55-1\implies \boxed{\bf n = 54}$

==pav38==

Answer 2

Răspuns:

2+2²+2³+....+2ⁿ=2⁵⁵-2

S=2+2²+2³+....+2ⁿ

2S=2(2¹+2²+2³+....+2ⁿ)

2S=2²+2³+2⁴+....+2ⁿ⁺¹

2S=(2¹+2²+2³+2⁴+....+2ⁿ)+2ⁿ⁺¹-2

2S=S+2ⁿ⁺¹-2

2S-S=2ⁿ⁺¹-2

S=2ⁿ⁺¹-2=>2⁵⁵=2ⁿ⁺¹=>55=n+1=>55-1=n=>54=n