Question

Cum se rezolva suma lui gauss la puteri

Answer 1

Exemplu:

1. Sa se calculeze 2¹+2²+2³+...+2ⁿ

Notam S=2¹+2²+2³+...+2ⁿ

S=2¹+2²+2³+...+2ⁿ    |×2

2S=2²+2³+2⁴+...+2ⁿ+2ⁿ⁺¹      |+2

2S+2=2+2²+2³+2⁴+...+2ⁿ+2ⁿ⁺¹

2S+2=S+2ⁿ⁺¹

S=2ⁿ⁺¹-2

2. Sa se calculeze 3¹+3²+3³+...+3ⁿ

Notam S= 3¹+3²+3³+...+3ⁿ   | ×3

3S=3²+3²+3⁴+...+3ⁿ+3ⁿ⁺¹  |+3

3S+3=3+3²+3²+3⁴+...+3ⁿ+3ⁿ⁺¹

3S+3=S+3ⁿ⁺¹

2S=3ⁿ⁺¹-3

$S=\frac{3^{n+1}-3}{2}$

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