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A-1-2-2-4-3-6-14-28-B-1-3-2-3-14-29-B-




Question Number 10340 by konen last updated on 04/Feb/17
A=1×2 +2×4 +3×6+...+14×28  B=1×3 +2×3 +...+14×29  ⇒B=?
$$\mathrm{A}=\mathrm{1}×\mathrm{2}\:+\mathrm{2}×\mathrm{4}\:+\mathrm{3}×\mathrm{6}+…+\mathrm{14}×\mathrm{28} \\ $$$$\mathrm{B}=\mathrm{1}×\mathrm{3}\:+\mathrm{2}×\mathrm{3}\:+…+\mathrm{14}×\mathrm{29} \\ $$$$\Rightarrow\mathrm{B}=? \\ $$
Commented by mrW1 last updated on 04/Feb/17
the definition of B is not clear.    do you mean   B=1×3 +2×5 +...+14×29 ?    if this is true, then  B=Σ_(n=1) ^(14) b_n =Σ_(n=1) ^(14) n(2n+1)  =2Σ_(n=1) ^(14) n^2 +Σ_(n=1) ^(14) n  =2×((14×(14+1)×(2×14+1))/6)+((14×(14+1))/2)  =((2×14×15×29)/6)+((14×15)/2)  =2030+105=2135
$${the}\:{definition}\:{of}\:{B}\:{is}\:{not}\:{clear}. \\ $$$$ \\ $$$${do}\:{you}\:{mean}\: \\ $$$$\mathrm{B}=\mathrm{1}×\mathrm{3}\:+\mathrm{2}×\mathrm{5}\:+…+\mathrm{14}×\mathrm{29}\:? \\ $$$$ \\ $$$${if}\:{this}\:{is}\:{true},\:{then} \\ $$$${B}=\underset{{n}=\mathrm{1}} {\overset{\mathrm{14}} {\sum}}{b}_{{n}} =\underset{{n}=\mathrm{1}} {\overset{\mathrm{14}} {\sum}}{n}\left(\mathrm{2}{n}+\mathrm{1}\right) \\ $$$$=\mathrm{2}\underset{{n}=\mathrm{1}} {\overset{\mathrm{14}} {\sum}}{n}^{\mathrm{2}} +\underset{{n}=\mathrm{1}} {\overset{\mathrm{14}} {\sum}}{n} \\ $$$$=\mathrm{2}×\frac{\mathrm{14}×\left(\mathrm{14}+\mathrm{1}\right)×\left(\mathrm{2}×\mathrm{14}+\mathrm{1}\right)}{\mathrm{6}}+\frac{\mathrm{14}×\left(\mathrm{14}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$=\frac{\mathrm{2}×\mathrm{14}×\mathrm{15}×\mathrm{29}}{\mathrm{6}}+\frac{\mathrm{14}×\mathrm{15}}{\mathrm{2}} \\ $$$$=\mathrm{2030}+\mathrm{105}=\mathrm{2135} \\ $$

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