Đề bài hỏi chứng minh hay làm gì hả bạn?

S=5+5²+5³+...+5²⁰²⁰+5²⁰²¹

5S=5²+5³+5⁴+...+5²⁰²²

5S-S=(5+5²+5³+...+5²⁰²⁰+5²⁰²¹)-(5²+5³+5⁴+...+5²⁰²²)

4S=5-5²⁰²²

S=(5-5²⁰²²):4

Ta có: \(\frac{1}{2^2}< \frac{1}{1.2}=\frac{1}{1}-\frac{1}{2}\); \(\frac{1}{3^2}< \frac{1}{2.3}=\frac{1}{2}-\frac{1}{3}\); ...; \(\frac{1}{100^2}< \frac{1}{99.100}=\frac{1}{99}-\frac{1}{100}\)

=> S < \(5\left(1-\frac{1}{100}\right)=5.\frac{99}{100}< 5.1=5\)=> S<5

Lại có: \(\frac{1}{2^2}>\frac{1}{2.3}=\frac{1}{2}-\frac{1}{3}\); \(\frac{1}{3^2}>\frac{1}{3.4}=\frac{1}{3}-\frac{1}{4}\); \(\frac{1}{100^2}>\frac{1}{100.101}=\frac{1}{100}-\frac{1}{101}\)

=> \(S>5\left(\frac{1}{2}-\frac{1}{101}\right)=5.\frac{101-2}{2.101}=\frac{5.99}{2.101}~2,45\)=> S>2

Vậy 2 < S < 5 => Đpcm

1) + S = 5 + 5² + 5³ + ... + 5⁹⁶ (có 96 số; 96 chia hết cho 6)

S = (5 + 5² + 5³ + 5⁴ + 5⁵ + 5⁶) + (5⁷ + 5⁸ + 5⁹ + 5¹⁰ + 5¹¹ + 5¹²) + ... + (5⁹¹ + 5⁹² + 5⁹³ + 5⁹⁴ + 5⁹⁵ + 5⁹⁶)

S = (5 + 5⁴) + (5² + 5⁵) + (5³ + 5⁶) + (5⁷ + 5¹⁰) + ... + (5⁹³ + 5⁹⁶)

S = 5.(1 + 5³) + 5².(1 + 5²) + 5³.(1 + 5³) + 5⁷.(1 + 5³) +  ... + 5⁹³.(1 + 5³)

S = 5.126 + 5².126 + 5³.126 + 5⁷.126 + ... + 5⁹³.126

S = 126.(5 + 5² + 5³ + 5⁷ + ... + 5⁹³) chia hết cho 126

+ Do 5 + 5² + 5³ + 5⁷ + ... + 5⁹³ chia hết cho 5 mà 126 chia hết cho 2

=> S chia hết cho 10 => S có tận cùng là 0

2) 16²⁰⁰⁸ - 8²⁰⁰⁰

= (...6) - (8⁴)⁵⁰⁰

= (...6) - (...6)⁵⁰⁰

= (...6) - (...6)

= (...0) chia hết cho 10

3) 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ + 10³ = (x + 1²)²

=> 1 + 8 + 27 + 64 + 125 + 216 + 343 + 512 + 729 + 1000 = (x + 1)²

=> (1 + 729) + (8 + 512) + (27 + 343) + (64 + 216) + 125 + 1000 = (x + 1)²

=> 730 + 520 + 370 + 280 + 1125 = (x + 1)²

=> (730 + 370) + (520 + 280) + 1125 = (x + 1)²

=> 1100 + 800 + 1125 = (x + 1)² 

=> 3025 = (x + 1)², vô lí

1) + S = 5 + 5² + 5³ + ... + 5⁹⁶ (có 96 số; 96 chia hết cho 6)

S = (5 + 5² + 5³ + 5⁴ + 5⁵ + 5⁶) + (5⁷ + 5⁸ + 5⁹ + 5¹⁰ + 5¹¹ + 5¹²) + ... + (5⁹¹ + 5⁹² + 5⁹³ + 5⁹⁴ + 5⁹⁵ + 5⁹⁶)

S = (5 + 5⁴) + (5² + 5⁵) + (5³ + 5⁶) + (5⁷ + 5¹⁰) + ... + (5⁹³ + 5⁹⁶)

S = 5.(1 + 5³) + 5².(1 + 5²) + 5³.(1 + 5³) + 5⁷.(1 + 5³) +  ... + 5⁹³.(1 + 5³)

S = 5.126 + 5².126 + 5³.126 + 5⁷.126 + ... + 5⁹³.126

S = 126.(5 + 5² + 5³ + 5⁷ + ... + 5⁹³) chia hết cho 126

+ Do 5 + 5² + 5³ + 5⁷ + ... + 5⁹³ chia hết cho 5 mà 126 chia hết cho 2

=> S chia hết cho 10 => S có tận cùng là 0

S=(6+5¹+5²+5³+.........5²⁰²⁰)x20

S=20x(5¹+5²)+20x(5³+5⁴)+...........20x(5²⁰¹⁹+5²⁰²⁰)+20x6

S=20x30+20x(5³+5⁴)+20x6+.........+20x(5²⁰¹⁹+5²⁰²⁰)

S=600+120+20x(5³+5⁴)...........+20x(5²⁰¹⁹+5²⁰²⁰)

Ta có:600+120+20x(5³+5⁴)+.........+20x(5²⁰¹⁹+5²⁰²⁰):hết cho 120

Vì 600:hết cho 120;120:hết cho 120;20x(5³+5⁴)+.............+20x(5²⁰¹⁹+5²⁰²⁰):hết cho 120

Nên S : hết cho 120

\(S=5+5^2+5^3+...+5^{1992}\)

\(=5\left(1+5\right)+5^3\left(1+5\right)+...+5^{1991}\left(1+5\right)\)

\(=5.6+5^3.6+...+5^{1991}.6=6\left(5+5^3+...+5^{1991}\right)⋮6\)

\(C=4+4^2+4^3+...+4^{2021}+4^{2022}\)

\(=\left(4+4^2\right)+\left(4^3+4^4\right)+...+\left(4^{2021}+4^{2022}\right)\)

\(=4.\left(1+4\right)+4^3.\left(1+4\right)+...+4^{2021}.\left(1+4\right)\)

\(=4.5+4^3.5+...+4^{2021}.5\)

\(=5.\left(4+4^3+...+4^{2021}\right)⋮5\)

Vậy \(C⋮5\)

Thách ai giải được hihihi

Đặt : \(A=5+5^2+5^3+...+5^{30}\)

\(=\left(5+5^2\right)+\left(5^3+5^4\right)+...+\left(5^{29}+5^{30}\right)\)

\(=5\left(1+5\right)+5^3\left(1+5\right)+...+5^{29}\left(1+5\right)\)

\(=\left(1+5\right)\left(5+5^3+...+5^{29}\right)\)

\(=6\left(5+5^3+...+5^{29}\right)⋮6\) (đpcm)

                                                   Bài giải

\(5+5^2+5^3+5^4+...+5^{29}+5^{30}\)

\(=\left(5+5^2\right)+\left(5^3+5^4\right)+...+\left(5^{29}+5^{30}\right)\)

\(=5\left(1+5\right)+5^3\left(1+5\right)+...+5^{29}\left(1+5\right)\)

\(=5\cdot6+5^3\cdot6+...+5^{29}\cdot6\)

\(=6\left(5+5^3+...+5^{29}\right)\text{ }⋮\text{ }6\)

\(\Rightarrow\text{ ĐPCM}\)