Question

1+3+3²+3³+3⁴+...+3⁹⁹
5+5²+5³+5⁴+...+5¹⁰⁰⁰
8+8²+8³+8⁴+...+8¹⁰³²
1+9+9²+9³+9⁴+...+9¹⁵⁰​

Answer 1

Răspuns:

Explicație pas cu pas:

a) S₁ = 1+3+3²+3³+3⁴+...+3⁹⁹    I ·3  =>

3·S₁ = 3+3²+3³+3⁴+...+3⁹⁹ +3¹⁰⁰ = S₁-1+3¹⁰⁰ =>

3S₁-S₁ = 3¹⁰⁰-1  => 2S₁ = 3¹⁰⁰-1 => S₁ = (3¹⁰⁰-1)/2

b) S₂ = 5+5²+5³+5⁴+...+5¹⁰⁰⁰   I·5  =>

5S₂ = 5²+5³+5⁴+...+5¹⁰⁰⁰+5¹⁰⁰¹ = S₂+5¹⁰⁰¹-5 =>

5S₂-S₂ = 5¹⁰⁰¹-5 => 4S₂ = 5¹⁰⁰¹-5 => S₂ = 5(5¹⁰⁰⁰-1)/4

c) S₃ = 8+8²+8³+8⁴+...+8¹⁰³²   I·8  =>

8S₃ = 8²+8³+8⁴+...+8¹⁰³²+8¹⁰³³ = S₃+8¹⁰³³-8 =>

7S₃ = 8¹⁰³³-8 => S₃ = 8(8¹⁰³²-1)/7

d) S₄ = 1+9+9²+9³+9⁴+...+9¹⁵⁰​  I·9  =>

9S₄ = 9+9²+9³+9⁴+...+9¹⁵⁰+9¹⁵¹ = S₄-1+9¹⁵¹ =>

8S⁴ = 9¹⁵¹-1  => S₄ = (9¹⁵¹-1)/8