a) \(S=1^5+3^5+....+75^5+99^5\)

\(\left(2a+1\right)^5-\left(2a+1\right)=2a\left(2a+1\right)\left(2a+2\right)\left[\left(2a+1\right)^2+1\right]\)

\(\left(2a+1\right)^5-\left(2a+1\right)=4a\left(2a+1\right)\left(a+1\right)\left[\left(2a+1\right)^2+1\right]⋮4\)

\(S=\left(1^5-1\right)+\left(3^5-3\right)+....+\left(75^5-75\right)+\left(99^5-99\right)+\left(1+3+5+...+75+99\right)\)

\(\Leftrightarrow\begin{matrix}1^5-1⋮4\\3^5-3⋮4\\5^5-5⋮4\\...........\\75^5-5⋮4\\99^5-99⋮4\end{matrix}\)

\(S_1=1+3+5+7+...+75+99=\frac{\left(1+75\right)\left[\frac{75-1}{2}+1\right]}{2}+99=38.38+96+3\)

\(\Rightarrow S_1:4\) dư 3

\(\Leftrightarrow S\) chia 4 dư 3

Sửa đề

\(\dfrac{3}{35}+\dfrac{3}{63}+\dfrac{3}{99}+...+\dfrac{3}{x\left(x+2\right)}=\dfrac{24}{35}\)

\(\dfrac{3}{5.7}+\dfrac{3}{7.9}+\dfrac{3}{9.11}+...+\dfrac{3}{x\left(x+2\right)}=\dfrac{24}{35}\)

\(\dfrac{3}{2}.\left(\dfrac{2}{5.7}+\dfrac{2}{7.9}+...+\dfrac{2}{x\left(x+2\right)}\right)=\dfrac{24}{35}\)

\(\dfrac{3}{2}.\left(\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{x}-\dfrac{1}{x+2}\right)=\dfrac{24}{35}\)

\(\dfrac{3}{2}.\left(\dfrac{1}{5}-\dfrac{1}{x+2}\right)=\dfrac{24}{35}\)

\(\dfrac{1}{5}-\dfrac{1}{x+2}=\dfrac{24}{35}:\dfrac{3}{2}\)

\(\dfrac{1}{5}-\dfrac{1}{x+2}=\dfrac{16}{35}\)

\(\dfrac{1}{x+2}=\dfrac{1}{5}-\dfrac{16}{35}\)

\(\dfrac{1}{x+2}=\dfrac{-9}{35}\)

\(x+2=35\)

\(x=35-2\)

\(x=33\)

\(-17x=-17-\left(-34\right)\)

\(-17x=-17\)

\(x=-1\)

Bạn coi lại đề: $\frac{3}{69}$ không nằm trong các số hạng có quy luật.

`@` `\text {Ans}`

`\downarrow`

`a,`

\(x \div 0,1 + x \div 0,25 - 35 = 35\)

`\Rightarrow x * 10 + x * 4 - 35 = 35`

`\Rightarrow x*(10+4) - 35 = 35`

`\Rightarrow 14x - 35 = 35`

`\Rightarrow 14x = 70`

`\Rightarrow x=70/14 =5`

Vậy, `x=5`

`b,`

\(x \div 0,125 - x \div 0,5 = 108\)

`\Rightarrow x * 4/5 - x*2=108`

`\Rightarrow x*(4/5 - 2) = 108`

`\Rightarrow -6/5x = 108`

`\Rightarrow x=108 \div -6/5`

`\Rightarrow x=-90`

Vậy, `x=-90.`

a) x/0,1 + x/0,25 - 35 = 35

x . 10 + x . 4 = 35 + 35

14x = 70

x = 70 : 14

x = 5

cảm ơn nhé

=00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

=00000000000000000000000000000000000000000000

3558.8 Tớ ko tính máy tính mà nó có 1 quy luật

3534.3  +  24.5

        3558.8

k nhé

3558,8

A = 99 x2 + 33+333+3333+...+333...33 (100 chữ số 3) 

đặt B = 33+333+3333+...+3333....33 = (99+999+9999+....+999...99): 3

= (10²-1+10³-1+10⁴-1+....+ 10¹⁰⁰-1):3

=  (10²+10³+10⁴+....+ 10¹⁰⁰-99):3

đặt  C= 10²+10³+10⁴+....+ 10¹⁰⁰

Cx 10 = 10³+10⁴+....+ 10¹⁰⁰+10¹⁰¹

^{Suy ra C x9 =} 10¹⁰¹- 10²

^{Bạn tính tiếp nhé.}

Ta có : \(\frac{x-99}{5}+\frac{x-99}{15}+\frac{x-99}{25}+\frac{x-99}{35}=0\)

\(\Rightarrow\left(x-99\right)\left(\frac{1}{5}+\frac{1}{15}+\frac{1}{25}+\frac{1}{35}\right)=0\)

Vì \(\frac{1}{5}+\frac{1}{15}+\frac{1}{25}+\frac{1}{35}\ne0\)

Nên : x - 99 = 0

Suy ra : x = 99

Vậy x = 99

(4+11):15+(35+64):99+(2+5):7=15:15+99:99+7:7=1+1+1=3

`4/15+35/99+11/15-2/7+64/99-5/7`

`=(4/15+11/15)+(35/99+64/99)-(2/7+5/7)`

`=15/15+99/99-7/7`

`=1+1-1`

`=2-1`

`=1`

1