\displaystyle-\frac{{4301}}{{100}} Explanation: \displaystyle-{3.45}-{2.3}{\left({8.6}\right)}\div{0.5} Rearrange! \displaystyle-\frac{{3.45}}{{100}}-\frac{{2.3}}{{10}}{\left(\frac{{8.6}}{{10}}\right)}\div\frac{{0.5}}{{10}} ...

\displaystyle{50},{000},{000} Explanation: When we are faced with a division by a decimal, change the denominator to a whole number. \displaystyle\frac{{{50},{000}}}{{0.001}}\times\frac{{1000}}{{1000}}=\frac{{{50},{000},{000}}}{{1}} ...

\displaystyle{0} Explanation: Count the number of terms first. In this case there are 4 terms. You can simplify within each term until there is a single answer for each term. Do more than one ...

\displaystyle-{352} Explanation: \displaystyle{\left(-{380}\right)}+{38}+{380}\div{\left(-{38}\right)} Use PEMDAS : \displaystyle-{380}+{38}+{380}\div-{38} Division to be carried out ...

\displaystyle{47} Explanation: To find an approximation while keeping the arithmetic a little easier, divide \displaystyle{705} by \displaystyle{15} : \displaystyle{705}={600}+{105}={\left({40}\cdot{15}\right)}+{\left({7}\cdot{15}\right)}={47}\cdot{15} ...

As everything is positive here a_1<a_2\iff a_1<\sqrt{a_1b_1}\iff a^2_1<a_1b_1\iff a_1(b_1-a_1)>0\;\color{red}\checkmark Do something similar for \;b_n\;