\(A=\dfrac{1}{2^0.5}+\dfrac{1}{2^1.5}+\dfrac{1}{2^2.5}+...+\dfrac{1}{2^8.5}\)

\(5A=\dfrac{1}{2^0}+\dfrac{1}{2^1}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^8}\)

\(5A=2-1+1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{4}+...++\dfrac{1}{128}+\dfrac{1}{256}\)

\(5A=2-\dfrac{1}{256}=\dfrac{511}{256}\)

\(A=\dfrac{511}{1280}\)

1/5 + 1/5  - 1/10 + 1/10 - 1/20 + 1/20 - 1/40 + ... + 1/640 - 1/1280

= 1/5 + 1/5 - 1/1280 = 511/1280

Diện tích hình thang là: 

\(\dfrac{\left(13+17\right).15}{2}=225\) cm vuông

\(\left\{10^2-\left[3^2.5+2^3.\left(4+2^{12}:2^{11}\right)\right]\right\}.2\)

\(=\left\{100-\left[3^2.5+2^3.\left(4+2\right)\right]\right\}.2\)

\(=\left\{100-\left[9.5+8.6\right]\right\}.2\)

\(=\left\{100-93\right\}.2\)

\(=7.2=14\)

\(\left\{10^2-\left[3^2\times5+2^3\times\left(4+2^{12}:2^{11}\right)\right]\right\}\times2\)

\(=\left\{100-\left[9\times5+8\times\left(4+2\right)\right]\right\}\times2\)

\(=\left\{100-\left[9\times5+8\times6\right]\right\}\times2\)

\(=\left\{100-93\right\}\times2\)

\(=7\times2\)

\(=14\)

\(B=8+8^3+8^5+...+8^{99}\\ \\ 64B=8^3+8^5+8^7+...+8^{101}\\ \\ \Rightarrow64B-B=8^{101}-8\\ \\ 63B=8^{101}-8\\ \\ \Rightarrow B=\dfrac{8^{101}-8}{63}\)

\(B=8+8^3+8^5+8^7+...+8^{99}\)

\(\Rightarrow B=8\left(1+8^2+8^4+8^6+...+8^{98}\right)\)

\(\Rightarrow B+8B=8.\left(8+8^3+8^5+8^7+...+8^{99}\right)+8\left(1+8^2+8^4+8^6+...+8^{98}\right)\)

\(\Rightarrow9B=8\left(1+8^1+8^2+8^3+8^4+...+8^{99}\right)\)

\(\Rightarrow9B=8.\dfrac{8^{99+1}}{8-1}-1\)

\(\Rightarrow B=8.\dfrac{8^{99+1}-1}{7.9}=\dfrac{8^{101}-8}{63}\)

\(\left\{{}\begin{matrix}a+b=221\\UCLN\left(a;b\right)=13\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=13m\\b=13n\\\left(m;n\right)=1\end{matrix}\right.\)

\(\Rightarrow13m+13n=221\)

\(\Rightarrow13\left(m+n\right)=221\)

\(\Rightarrow m+n=17\)

- Với \(\left\{{}\begin{matrix}m=16\\n=1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=208\\b=13\end{matrix}\right.\)

- Với \(\left\{{}\begin{matrix}m=14\\n=3\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=182\\b=39\end{matrix}\right.\)

- Với \(\left\{{}\begin{matrix}m=12\\n=5\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=156\\b=65\end{matrix}\right.\)

- Với \(\left\{{}\begin{matrix}m=10\\n=7\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=130\\b=91\end{matrix}\right.\)

- Với \(\left\{{}\begin{matrix}m=6\\n=11\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=78\\b=143\end{matrix}\right.\)

\(\Rightarrow\left(a;b\right)\in\left\{\left(108;13\right);\left(182;39\right);\left(156:65\right);\left(130;91\right);\left(78;143\right)\right\}\)

Vận tốc lúc đi so với vận tốc lúc về bằng: 10/12= 5/6

Tổng số phần bằng nhau: 5+6=11 (phần)

Đổi 3h40p=11/3h

Thời gian về bằng:

11/3 : 11 x 5 = 5/3(h)

Quãng đường AB dài:

5/3 x 12= 20(km)

Đ.số: 20km

\(\dfrac{3}{7}+\left(-\dfrac{5}{2}\right)+\left(-\dfrac{3}{5}\right)=\dfrac{30}{70}+\left(-\dfrac{175}{70}\right)+\left(-\dfrac{42}{70}\right)=-\dfrac{187}{70}\)

\(\dfrac{3}{7}+\left(-\dfrac{5}{2}\right)+\left(-\dfrac{3}{5}\right)=\dfrac{3}{7}-\dfrac{5}{2}-\dfrac{3}{5}\)

\(=\dfrac{30}{70}-\dfrac{175}{70}-\dfrac{42}{70}=-\dfrac{187}{70}\)

a) x=0

b) x=0

a) \(BCNN\left(3;8\right)=24\)

\(\Rightarrow x=24\) thỏa đề bài

b) \(BCNN\left(12;18;30\right)=180\)

\(\Rightarrow x=180\) thỏa đề bài

a) Ta có 111 chia hết cho 37 mà các số dạng aaa khi nào cũng chia hết cho 111 ⇒ Các số có dạng aaa luôn chia hết cho 37 (ĐPCM)

b) Ta có ab-ba=a.10+b-b.10-a=9.a-9.b=9.(a-b)

      Vì 9 chia hết cho 9 ⇒ 9.(a-b) chia hết cho 9 ⇒ ab-ba bao giờ cũng chia hết cho 9 (ĐPCM)

c) Ta có 2 trường hợp n có hạng 2k hoặc 2k+1

+) Nếu n= 2k thì n+6 chia hết cho 2 ⇒ (n+3)(n+6) chia hết cho 2

+) Nếu n= 2k+1 thì n+3 chia hết cho 2 ⇒ (n+3)(n+6) chia hết cho 2

 ⇒ (n+3)(n+6) chia hết cho 2 với mọi n là số tự nhiên

a) \(\overline{aaa}=100a+10a+a=111a\)

mà \(111=37.3⋮37\)

\(\Rightarrow\overline{aaa}⋮37\left(dpcm\right)\)

b) \(\overline{ab}-\overline{ba}=10a+b-10b-a=9a-9b=9\left(a-b\right)⋮9\left(a\ge b\right)\)

\(\Rightarrow dpcm\)

\(\left[\left(2,18-1,38\right).1,25\right]:\left[\left(17,9-15,65\right):1,125\right]:\left(-0,4\right)\)

\(=\left[0,18.1,25\right]:\left[2,25:1,125\right]:\left(-0,4\right)\)

\(=\left[0,225\right]:\left[2\right]:\left(-0,4\right)\)

\(=\left[0,225\right]:\left[2\right]:\left(-0,4\right)=-0,045\)

[ (2,18 - 1,38).1,25]: [(17,9 - 15,65): 1,125]:(-0,4)

= [0,8.1,25]: [(2,25): 1,125]:(-0,4)

= 1:[ (2,25): 1,125]:(-0,4)

= 1:(2):(-0,4)

= 0,5: -(0,4)

= - \(\dfrac{5}{4}\)