Câu 1:
\(-\frac{1}{54}-\frac{3}{2}\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{79.81}\right)\)
\(=-\frac{1}{54}-\frac{3}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{79}-\frac{1}{81}\right)\)
\(=-\frac{1}{54}-\frac{3}{2}\left(1-\frac{1}{81}\right)\)
\(=-\frac{1}{54}-\frac{40}{27}\)
\(=-\frac{3}{2}\)
Câu 2:
\(a^2+b^2+c^2+d^2+e^2=\left(a+b+c+d+e\right)^2-2\left(ab+ac+ad+ae+bc+bd+be+cd+ce+de\right)\)
Mà \(2\left(ab+ac+ad+ae+bc+bd+be+cd+ce+de\right)⋮2\)
\(\Rightarrow\left(a+b+c+d+e\right)^2⋮2\)
\(\Rightarrow a+b+c+d+e⋮2\)
Do \(a,b,c,d,e\) nguyên dương \(\Rightarrow a+b+c+d+e>2\Rightarrow a+b+c+d+e\) là hợp số
Câu 3:
- Chiều thuận: \(3a+2b⋮17\Rightarrow10a+b⋮17\)
Ta có \(\left\{{}\begin{matrix}17a⋮17\\3a+2b⋮17\end{matrix}\right.\) \(\Rightarrow17a+3a+2b⋮17\Rightarrow20a+2b⋮17\)
\(\Rightarrow2\left(10a+b\right)⋮17\), mà 2 và 17 nguyên tố cùng nhau \(\Rightarrow10a+b⋮17\)
- Chiều nghịch: \(10a+b⋮17\Rightarrow3a+2b⋮17\)
\(10a+b⋮17\Rightarrow2\left(10a+b\right)⋮17\Rightarrow20a+2b⋮17\)
\(\Rightarrow17a+3a+2b⋮17\)
Mà \(17a⋮17\Rightarrow3a+2b⋮17\) (đpcm)