Chứng minh :
$\frac{1}{3}$ + $\frac{1}{7}$ + $\frac{1}{13}$ + $\frac{1}{21}$ +$\frac{1}{31}$ + $\frac{1}{43}$ + $\frac{1}{57}$ + $\frac{1}{73}$ + $\frac{1}{91}$ < 1
Chứng minh :
$\frac{1}{3}$ + $\frac{1}{7}$ + $\frac{1}{13}$ + $\frac{1}{21}$ +$\frac{1}{31}$ + $\frac{1}{43}$ + $\frac{1}{57}$ + $\frac{1}{73}$ + $\frac{1}{91}$ < 1
Đáp án:
`A=1/3+1/7+1/13+1/21+1/31+1/43+1/57+1/73+1/91`
`=>A=1/(1xx2+1)+1/(2xx3+1)+1/(3xx4+1)+1/(4xx5+1)+1/(5xx6+1)+1/(6xx7+1)+1/(7xx8+1)+1/(8xx9+1)+1/(9xx10+1`
`=>A<1/(1xx2)+1/(2xx3)+1/(3xx4)+1/(4xx5)+1/(5xx6)+1/(6xx7)+1/(7xx8)+1/(8xx9)+1/(9xx10`
`=>A<1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10`
`=>A<1-1/10<1`
`=>A<1`
Vậy biểu thức có giá trị bé hơn 1.
Gọi $\frac{1}{3}$ + $\frac{1}{7}$ + $\frac{1}{13}$ +…+ $\frac{1}{91}$ = A
Ta có :
$\frac{1}{3}$ < $\frac{1}{1.2}$
$\frac{1}{7}$ < $\frac{1}{2.3}$
$\frac{1}{13}$ < $\frac{1}{3.4}$
………….
………….
$\frac{1}{91}$ < $\frac{1}{9.10}$
⇒A < $\frac{1}{1.2}$ + $\frac{1}{2.3}$ + $\frac{1}{3.4}$ + … + $\frac{1}{9.10}$
⇒A < 1 – $\frac{1}{2}$ + $\frac{1}{2}$ – $\frac{1}{3}$ + $\frac{1}{3}$ – $\frac{1}{4}$ + … + $\frac{1}{9}$ – $\frac{1}{10}$
⇒A < 1 – $\frac{1}{10}$
⇒A < $\frac{9}{10}$
Mà $\frac{9}{10}$ < 1 ⇒ A < 1
⇒ ĐPCM