Cho A=1/2^2+1/3^2 +1/4^2+…+1/150^2 CMR :A<1 13/10/2021 Bởi Melody Cho A=1/2^2+1/3^2 +1/4^2+…+1/150^2 CMR :A<1
Đáp án: $\begin{array}{l}{2^2} > 1.2 \Rightarrow \dfrac{1}{{{2^2}}} < \dfrac{1}{{1.2}}\\ \Rightarrow \dfrac{1}{{{3^2}}} < \dfrac{1}{{2.3}};\dfrac{1}{{{4^2}}} < \dfrac{1}{{3.4}};…;\dfrac{1}{{{{150}^2}}} < \dfrac{1}{{149.150}}\\ \Rightarrow \dfrac{1}{{{2^2}}} + \dfrac{1}{{{3^2}}} + .. + \dfrac{1}{{{{150}^2}}} < \dfrac{1}{{1.2}} + \dfrac{1}{{2.3}} + \dfrac{1}{{3.4}} + .. + \dfrac{1}{{149.150}}\\ \Rightarrow A < 1 – \dfrac{1}{2} + \dfrac{1}{2} – \dfrac{1}{3} + \dfrac{1}{3} – \dfrac{1}{4} + … + \dfrac{1}{{149}} – \dfrac{1}{{150}}\\ \Rightarrow A < 1 – \dfrac{1}{{150}} < 1\\Vậy\,A < 1\end{array}$ Bình luận
Ta có S<$\frac{1}{1.2}$+ $\frac{1}{2.3}$+…+ $\frac{1}{149.150}$ ⇒S<1-$\frac{1}{2}$ $\frac{1}{2}$ +….+$\frac{1}{-149}$+$\frac{1}{149}$- $\frac{1}{150}$ ⇒S<1-$\frac{1}{150}$ Vì 1-$\frac{1}{150}$ <1 ⇒S<1 Bình luận
Đáp án:
$\begin{array}{l}
{2^2} > 1.2 \Rightarrow \dfrac{1}{{{2^2}}} < \dfrac{1}{{1.2}}\\
\Rightarrow \dfrac{1}{{{3^2}}} < \dfrac{1}{{2.3}};\dfrac{1}{{{4^2}}} < \dfrac{1}{{3.4}};…;\dfrac{1}{{{{150}^2}}} < \dfrac{1}{{149.150}}\\
\Rightarrow \dfrac{1}{{{2^2}}} + \dfrac{1}{{{3^2}}} + .. + \dfrac{1}{{{{150}^2}}} < \dfrac{1}{{1.2}} + \dfrac{1}{{2.3}} + \dfrac{1}{{3.4}} + .. + \dfrac{1}{{149.150}}\\
\Rightarrow A < 1 – \dfrac{1}{2} + \dfrac{1}{2} – \dfrac{1}{3} + \dfrac{1}{3} – \dfrac{1}{4} + … + \dfrac{1}{{149}} – \dfrac{1}{{150}}\\
\Rightarrow A < 1 – \dfrac{1}{{150}} < 1\\
Vậy\,A < 1
\end{array}$
Ta có S<$\frac{1}{1.2}$+ $\frac{1}{2.3}$+…+ $\frac{1}{149.150}$
⇒S<1-$\frac{1}{2}$ $\frac{1}{2}$ +….+$\frac{1}{-149}$+$\frac{1}{149}$- $\frac{1}{150}$
⇒S<1-$\frac{1}{150}$
Vì 1-$\frac{1}{150}$ <1
⇒S<1