Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 194135 by cortano12 last updated on 28/Jun/23

determinant (((((23!−23)/(1.1!+2.2!+3.3!+...+21.21!)) =?)))

$$\:\:\:\:\:\:\begin{array}{|c|}{\frac{\mathrm{23}!−\mathrm{23}}{\mathrm{1}.\mathrm{1}!+\mathrm{2}.\mathrm{2}!+\mathrm{3}.\mathrm{3}!+...+\mathrm{21}.\mathrm{21}!}\:=?}\\\hline\end{array} \\ $$

Answered by Frix last updated on 28/Jun/23

23 [Σ_(j=1) ^n j×j!=(n+1)!−1]

$$\mathrm{23} \\ $$$$\left[\underset{{j}=\mathrm{1}} {\overset{{n}} {\sum}}{j}×{j}!=\left({n}+\mathrm{1}\right)!−\mathrm{1}\right] \\ $$

Answered by aba last updated on 28/Jun/23

1.1!+2.2!+...+21.21!=(2−1)1!+(3−1)2!+...+(22−1)21! =2!−1!+3!−2!+...+22!−21! =22!−1! A=((23!−23)/(1.1!+2.2!+...+21.21!))=((23(22!−1))/(22!−1!))=23 ✓

$$\mathrm{1}.\mathrm{1}!+\mathrm{2}.\mathrm{2}!+...+\mathrm{21}.\mathrm{21}!=\left(\mathrm{2}−\mathrm{1}\right)\mathrm{1}!+\left(\mathrm{3}−\mathrm{1}\right)\mathrm{2}!+...+\left(\mathrm{22}−\mathrm{1}\right)\mathrm{21}! \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\cancel{\mathrm{2}!}−\mathrm{1}!+\cancel{\mathrm{3}!}−\cancel{\mathrm{2}!}+\cancel{.}.\cancel{.}+\mathrm{22}!−\cancel{\mathrm{21}!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{22}!−\mathrm{1}! \\ $$$$\mathrm{A}=\frac{\mathrm{23}!−\mathrm{23}}{\mathrm{1}.\mathrm{1}!+\mathrm{2}.\mathrm{2}!+...+\mathrm{21}.\mathrm{21}!}=\frac{\mathrm{23}\left(\mathrm{22}!−\mathrm{1}\right)}{\mathrm{22}!−\mathrm{1}!}=\mathrm{23}\:\checkmark\: \\ $$$$ \\ $$

Answered by MM42 last updated on 28/Jun/23

n×n!=(n+1)!−n! ⇒s=1×1!+2×2!+3×3!+...+21×22!= (2!−1!)+(3!−2!)+(4!−3!)+...+(21!−20!)+(22!−21!)= 22!−1 ⇒((23!−23)/s)=((23(22!−1))/(22!−1))=23 ✓

$${n}×{n}!=\left({n}+\mathrm{1}\right)!−{n}! \\ $$$$\Rightarrow{s}=\mathrm{1}×\mathrm{1}!+\mathrm{2}×\mathrm{2}!+\mathrm{3}×\mathrm{3}!+...+\mathrm{21}×\mathrm{22}!= \\ $$$$\left(\mathrm{2}!−\mathrm{1}!\right)+\left(\mathrm{3}!−\mathrm{2}!\right)+\left(\mathrm{4}!−\mathrm{3}!\right)+...+\left(\mathrm{21}!−\mathrm{20}!\right)+\left(\mathrm{22}!−\mathrm{21}!\right)= \\ $$$$\mathrm{22}!−\mathrm{1} \\ $$$$\Rightarrow\frac{\mathrm{23}!−\mathrm{23}}{{s}}=\frac{\mathrm{23}\left(\mathrm{22}!−\mathrm{1}\right)}{\mathrm{22}!−\mathrm{1}}=\mathrm{23}\:\checkmark \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com