Menu Close

The-coefficient-of-x-n-2-in-the-polynomial-x-1-x-2-x-n-is-




Question Number 8132 by 314159 last updated on 01/Oct/16
The  coefficient of x^(n−2)  in the polynomial  (x−1)(x−2)....(x−n)  is
$$\mathrm{The}\:\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{n}−\mathrm{2}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{polynomial} \\ $$$$\left({x}−\mathrm{1}\right)\left({x}−\mathrm{2}\right)….\left({x}−{n}\right)\:\:\mathrm{is} \\ $$
Commented by Yozzia last updated on 01/Oct/16
coefficient(x^(n−2) )=Σ_(∀α,β∈[1,n]) αβ  1×2+1×3+1×4+...+1×n  +2×3+2×4+2×5+...+2×n  +3×4+3×5+...+3+n  +...+(n−1)×n
$${coefficient}\left({x}^{{n}−\mathrm{2}} \right)=\underset{\forall\alpha,\beta\in\left[\mathrm{1},{n}\right]} {\sum}\alpha\beta \\ $$$$\mathrm{1}×\mathrm{2}+\mathrm{1}×\mathrm{3}+\mathrm{1}×\mathrm{4}+…+\mathrm{1}×{n} \\ $$$$+\mathrm{2}×\mathrm{3}+\mathrm{2}×\mathrm{4}+\mathrm{2}×\mathrm{5}+…+\mathrm{2}×{n} \\ $$$$+\mathrm{3}×\mathrm{4}+\mathrm{3}×\mathrm{5}+…+\mathrm{3}+{n} \\ $$$$+…+\left({n}−\mathrm{1}\right)×{n} \\ $$$$ \\ $$$$ \\ $$
Answered by prakash jain last updated on 01/Oct/16
From Yozzi′s comment  Σ_(j=1) ^(n−1)  Σ_(i=j+1) ^n j∙i  Please see post from sou Q8139 for correct answer.
$$\mathrm{From}\:\mathrm{Yozzi}'\mathrm{s}\:\mathrm{comment} \\ $$$$\underset{{j}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}\:\underset{{i}={j}+\mathrm{1}} {\overset{{n}} {\sum}}{j}\centerdot{i} \\ $$$$\mathrm{Please}\:\mathrm{see}\:\mathrm{post}\:\mathrm{from}\:\mathrm{sou}\:{Q}\mathrm{8139}\:\mathrm{for}\:\mathrm{correct}\:\mathrm{answer}. \\ $$
Commented by prakash jain last updated on 02/Oct/16
Σ_(j=1) ^(n−1) j(Σ_(i=1) ^n i−Σ_(i=1) ^j j)  =Σ_(j=1) ^(n−1) j((n(n+1))/2)−Σ_(j=1) ^(n−1) j((j(j+1))/2)  =((n(n+1)n(n−1))/4)−Σ_(j=1) ^(n−1) ((j^3 +j^2 )/2)  =((n^2 (n−1)(n+1))/4)−(1/2){(((n−1)n)/2)}^2 −(1/2){(((n−1)n(2n−1))/6)}  =((n(n−1))/(24))[6n(n+1)−3n(n−1)−2(2n−1)]  =((n(n−1))/(24))[6n^2 +6n−3n^2 +3n−4n+2]  =((n(n−1))/(24))[3n^2 +5n+2]  =((n(n−1))/(24))[3n^2 +3n+2n+2]  =((n(n−1)(n+1)(3n+2))/(24))
$$\underset{{j}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}{j}\left(\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{i}−\underset{{i}=\mathrm{1}} {\overset{{j}} {\sum}}{j}\right) \\ $$$$=\underset{{j}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}{j}\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}−\underset{{j}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}{j}\frac{{j}\left({j}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$=\frac{{n}\left({n}+\mathrm{1}\right){n}\left({n}−\mathrm{1}\right)}{\mathrm{4}}−\underset{{j}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}\frac{{j}^{\mathrm{3}} +{j}^{\mathrm{2}} }{\mathrm{2}} \\ $$$$=\frac{{n}^{\mathrm{2}} \left({n}−\mathrm{1}\right)\left({n}+\mathrm{1}\right)}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{2}}\left\{\frac{\left({n}−\mathrm{1}\right){n}}{\mathrm{2}}\right\}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}}\left\{\frac{\left({n}−\mathrm{1}\right){n}\left(\mathrm{2}{n}−\mathrm{1}\right)}{\mathrm{6}}\right\} \\ $$$$=\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{24}}\left[\mathrm{6}{n}\left({n}+\mathrm{1}\right)−\mathrm{3}{n}\left({n}−\mathrm{1}\right)−\mathrm{2}\left(\mathrm{2}{n}−\mathrm{1}\right)\right] \\ $$$$=\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{24}}\left[\mathrm{6}{n}^{\mathrm{2}} +\mathrm{6}{n}−\mathrm{3}{n}^{\mathrm{2}} +\mathrm{3}{n}−\mathrm{4}{n}+\mathrm{2}\right] \\ $$$$=\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{24}}\left[\mathrm{3}{n}^{\mathrm{2}} +\mathrm{5}{n}+\mathrm{2}\right] \\ $$$$=\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{24}}\left[\mathrm{3}{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{2}{n}+\mathrm{2}\right] \\ $$$$=\frac{{n}\left({n}−\mathrm{1}\right)\left({n}+\mathrm{1}\right)\left(\mathrm{3}{n}+\mathrm{2}\right)}{\mathrm{24}} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *