Question Number 10339 by konen last updated on 04/Feb/17
$$\mathrm{A}=\mathrm{1}×\mathrm{2}\:+\:\mathrm{2}×\mathrm{3}\:+\mathrm{3}×\mathrm{4}+...+\mathrm{10}×\mathrm{11} \\ $$$$\mathrm{B}=\mathrm{3}×\mathrm{8}\:+\mathrm{6}×\mathrm{12}\:+\mathrm{9}×\mathrm{16}+...+\mathrm{30}×\mathrm{44} \\ $$$$\Rightarrow\frac{\mathrm{A}}{\mathrm{B}}=? \\ $$
Answered by mrW1 last updated on 04/Feb/17
$${a}_{{n}} ={n}×\left({n}+\mathrm{1}\right) \\ $$$${b}_{{n}} =\mathrm{3}×{n}×\mathrm{4}×\left({n}+\mathrm{1}\right)=\mathrm{12}×{n}×\left({n}+\mathrm{1}\right)=\mathrm{12}{a}_{{n}} \\ $$$${A}=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}{a}_{{n}} \\ $$$${B}=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}{b}_{{n}} =\mathrm{12}\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}{a}_{{n}} =\mathrm{12}{A} \\ $$$$\Rightarrow\:\frac{{A}}{{B}}=\frac{{A}}{\mathrm{12}{A}}=\frac{\mathrm{1}}{\mathrm{12}} \\ $$
Answered by arge last updated on 05/Feb/17
$$ \\ $$$${A},\:\:{n}=\mathrm{10},\:{d}=\mathrm{4} \\ $$$${a}_{\mathrm{1}} =\mathrm{2} \\ $$$${a}_{\mathrm{2}} =\mathrm{6} \\ $$$${a}_{\mathrm{10}} =\mathrm{110} \\ $$$$ \\ $$$${S}_{{n}} =\frac{\left({a}_{{n}} −{a}_{\mathrm{1}} +{d}\right)\left({a}_{\mathrm{1}} +{a}_{{n}} \right)}{\mathrm{2}{d}} \\ $$$$ \\ $$$${S}_{{n}} {A}=\mathrm{1568} \\ $$$$ \\ $$$${B},\:{n}=?\:,\:{d}=\mathrm{48} \\ $$$${a}_{\mathrm{1}} =\mathrm{24} \\ $$$${a}_{\mathrm{2}} =\mathrm{72} \\ $$$${a}_{{n}} =\mathrm{1320} \\ $$$$ \\ $$$${n}=\frac{{a}_{{n}} −{a}_{\mathrm{1}} }{{d}}\:+\mathrm{1}=\mathrm{28} \\ $$$$ \\ $$$${S}_{{n}} {B}=\mathrm{32256} \\ $$$$ \\ $$$$ \\ $$$${A}/{B}=\frac{\mathrm{7}}{\mathrm{144}} \\ $$$$ \\ $$
Commented by mrW1 last updated on 05/Feb/17
$${I}\:{don}'{t}\:{think}\:{this}\:{is}\:{correct},\:{sir}.\:{Both} \\ $$$${A}\:{and}\:{B}\:{are}\:{not}\:{A}.{P}.! \\ $$
Commented by arge last updated on 05/Feb/17
$$ \\ $$$$ \\ $$$${La}\:{calculadora}\:{que}\:{ofrecio}\:{los}\:{datos}\:{para}\:{el}\:{lanzamiento}\:{del}\:{transbordador}\:{espacial}\:{en}\:{USA},\:{fallo}\:{y}\:{ocurrio}\:{el}\:{accidente}. \\ $$$$ \\ $$
Commented by mrW1 last updated on 06/Feb/17
$${can}\:{you}\:{please}\:{translate}\:{into}\:{english}? \\ $$