a)\(\frac{8}{9}.\frac{15}{16}.\frac{24}{25}.....\frac{2499}{2500}\) \(=\frac{2.4}{3^2}.\frac{3.5}{4^2}.\frac{4.6}{5^2}.....\frac{49.51}{50^2}\)\(=\frac{\left(2.3.4.....49\right)\left(4.5.6....51\right)}{\left(3.4.5.....50\right)\left(3.4.5.....50\right)}=\frac{2.51}{50.3}=\frac{1.17}{25}=\frac{17}{25}\)

b)\(\left(1-\frac{1}{3}\right)\left(1-\frac{1}{6}\right)\left(1-\frac{1}{10}\right)......\left(1-\frac{1}{780}\right)\)

\(=\frac{2}{3}.\frac{5}{6}.\frac{9}{10}......\frac{779}{780}\)\(=\frac{4}{6}.\frac{10}{12}.\frac{18}{20}.....\frac{1558}{1560}\)

\(=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}.....\frac{38.41}{39.40}\)

\(=\frac{\left(1.2.3.....38\right)\left(4.5.6.....41\right)}{\left(2.3.4.....39\right)\left(3.4.5.....40\right)}=\frac{1.41}{39.3}=\frac{41}{117}\)

\(\frac{4}{1}\)

\(\text{ta có: }\frac{1}{2.3}=\frac{1}{2}-\frac{1}{3}\)

             \(\frac{1}{3.4}=\frac{1}{3}-\frac{1}{4}\)

           \(\frac{1}{4.5}=\frac{1}{4}-\frac{1}{5}\)

           ...........................

             \(\frac{1}{39.40}=\frac{1}{39}-\frac{1}{40}\)

Đồng nhất 2 vế ta có:

   \(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+....+\frac{1}{39}-\frac{1}{40}=\frac{1}{2}-\frac{1}{40}=\frac{19}{40}\)

Ta có :

\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{39.40}\)

\(=\)\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{39}-\frac{1}{40}\)

\(=\)\(\frac{1}{2}-\frac{1}{40}\)

\(=\)\(\frac{20}{40}-\frac{1}{40}\)

\(=\)\(\frac{19}{40}\)

Vậy \(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{39.40}=\frac{19}{40}\)

\(S=1^2+2^2+3^2+...+30^2\)

<=>\(S=1\left(2-1\right)+2\left(3-1\right)+3\left(4-1\right)+...+30\left(31-1\right)\)

<=>\(S=1.2-1+2.3-2+3.4-3+...+30.31-30\)

<=>\(S=\left(1.2+2.3+3.4+...+30.11\right)-\left(1+2+3+...+30\right)\)

Đặt \(A=1.2+2.3+3.4+...+30.11\) và \(B=1+2+3+...+30\)

Thôi đến bước này bạn tự tính đi, dễ rồi, ghi hết thì ... mình mệt =))) 

được: \(A=\frac{30.31.32}{3}=10.31.32=9920;B=\frac{30.31}{2}=15.31=465\)

=>S=9920+465=10385 không phải số chính phương

Nhầm phần cuối, như này mới đúng này:"9920-465=9455 ko phải số chính phương" nhé

Ta có: \(\frac{1}{2}=1-\frac{1}{2}\);

\(\frac{1}{4}=\frac{1}{2}-\frac{1}{4}\); \(\frac{1}{8}=\frac{1}{4}-\frac{1}{8}\);...; \(\frac{1}{512}=\frac{1}{256}-\frac{1}{512}\); \(\frac{1}{1024}=\frac{1}{512}-\frac{1}{1024}\)

=> \(A=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{512}+\frac{1}{1024}\)

=> \(A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{8}+...+\frac{1}{256}-\frac{1}{512}+\frac{1}{512}-\frac{1}{1024}\)

=> \(A=1-\frac{1}{1024}=\frac{1023}{1024}\)

Đáp số: \(A=\frac{1023}{1024}\)

Việt Trì trong trái tim em dk

viết dấu có đc k ạ

viet co dau nha

a) Để â nhận giá trị nguyên

\(\Rightarrow8n-9⋮2n+5\)

\(\Rightarrow8n+20-29⋮2n+5\)

\(\Rightarrow4.\left(2n+5\right)-29⋮2n+5\)

mà \(4.\left(2n+5\right)⋮2n+5\)

\(\Rightarrow-29⋮2n+5\)

\(\Rightarrow2n+5\inƯ\left(-29\right)\)

tự làm nốt nhé, tick nha

khó quá!!!Bó tay...Sorry