bài 1 : Tính
1) \(\left(\frac{1}{2}+1\right).\left(\frac{1}{3}+1\right).\left(\frac{1}{4}+1\right).....\left(\frac{1}{99}+1\right)\)
2) \(\left(\frac{1}{2}-1\right).\left(\frac{1}{3}-1\right).\left(\frac{1}{4}-1\right).....\left(\frac{1}{99}-1\right)\)
3) \(\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}...........\frac{899}{30^2}\)
4) \(\frac{4}{3.7}+\frac{4}{7.11}+\frac{4}{11.15}+......+\frac{4}{107.111}\)
5) \(\frac{2}{15}+\frac{2}{35}+\frac{2}{63}+.........+\frac{2}{399}\)
6) \(\frac{7}{10.11}+\frac{7}{11.12}+\frac{7}{12.13}+....+\frac{7}{69.70}\)
7) \(\frac{6}{15.18}+\frac{6}{18.21}+\frac{6}{21.24}+.....+\frac{6}{87.90}\)
1) =3/2 . 4/3 . 5/4 ...... 100/99 = 100/2 = 50
2) = -1/2 . -2/3 .-3/4 ..... . -98/99 = 1/99 (Tích này có 98 thừa số âm, 98 là số chẵn nên tích mang dấu dương)
3) =(1.3)/(2.2) . (2.4)/(3.3) . (3.5)/(4.4) ......... (29.31)/(30.30)
= (1.2.3.....31)/(2.3.4......30) . (3.4.5......29)/(2.3.4....30)
= 31 . 1/60
= 31/60
4) =1/3 - 1/7 + 1/7 - 1/11 + 1/11 - 1/15 + ... + 1/107 - 1/111
= 1/3 - 1/111
= 12/37
5) = 2/(3.5) + 2/(5.7) + 2/(7.9) +... + 2/(19.21)
= 1/3 - 1/5 + 1/5 - 1/7 + 1/7 - 1/9 + .... + 1/19 - 1/21
= 1/3 - 1/21
= 2/7
6) Đặt biểu thức là A, lấy A chia cho 7 được
A/7 = 1/(10.11) + 1/(11.12) + 1/(12.13) + ... + 1/ (69.70)
= 1/10 - 1/11 + 1/11 - 1/12 + 1/12 - 1/13 + ... + 1/69 - 1/70
= 1/10 - 1/70
= 3/35
Suy ra A = 3/35 . 7 = 3/5
7) Đặt biểu thức là B, lấy B chia cho 2 được
B/2 = 3/(15.18) + 3/(18.21) + 3/(21.24) +... + 3/(87.90)
= 1/15 - 1/18 + 1/18 - 1/21 + 1/21 - 1/24 + ... + 1/87 - 1/90
= 1/15 - 1/90
= 1/18
Suy ra B = 1/18 . 2 = 1/9
\(G=\left(\frac{1}{2}+1\right)\left(\frac{1}{3}+1\right)\left(\frac{1}{4}+1\right)......\left(\frac{1}{99}+1\right)\)
\(=\frac{3}{2}\cdot\frac{4}{3}\cdot\frac{5}{4}\cdot......\cdot\frac{100}{99}\)
\(\frac{100}{2}=50\)
\(I=\left(\frac{1}{2}-1\right)\left(\frac{1}{3}-1\right)\left(\frac{1}{4}-1\right)....\left(\frac{1}{99}-1\right)\)
\(=\left(-\frac{1}{2}\right)\left(-\frac{2}{3}\right)\left(-\frac{3}{4}\right).....\left(-\frac{98}{99}\right)\)
Do số thừa số là số chẵn,nên:
\(I=\frac{1}{2}\cdot\frac{2}{3}\cdot.....\cdot\frac{98}{99}=\frac{1}{99}\)
\(A=\frac{3}{2^2}\cdot\frac{8}{3^2}\cdot\frac{15}{4^2}.....\cdot\frac{899}{30^2}\)
\(=\frac{3\cdot1}{2\cdot2}\cdot\frac{4\cdot2}{3\cdot3}\cdot\frac{3\cdot5}{4\cdot4}\cdot.....\cdot\frac{29\cdot31}{30\cdot30}\)
\(=\frac{1\cdot31}{2\cdot30}=\frac{31}{60}\)
\(GIA=\frac{4}{3\cdot7}+\frac{4}{7\cdot11}+\frac{4}{11\cdot15}+.....+\frac{4}{107\cdot111}\)
\(=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+\frac{1}{11}-\frac{1}{15}+.....+\frac{1}{107}-\frac{1}{111}\)
\(=\frac{1}{3}-\frac{1}{111}=\frac{108}{333}\)
