Question and Answers Forum

All Questions      Topic List

Permutation and Combination Questions

Previous in All Question      Next in All Question      

Previous in Permutation and Combination      Next in Permutation and Combination      

Question Number 86684 by john santu last updated on 30/Mar/20

If (1+px+qx^2 )^8 = 1+8x+52x^2 +kx^3 +... find p , q and k.

$$\mathrm{If}\:\left(\mathrm{1}+\mathrm{px}+\mathrm{qx}^{\mathrm{2}} \right)^{\mathrm{8}} \:=\:\mathrm{1}+\mathrm{8x}+\mathrm{52x}^{\mathrm{2}} +\mathrm{kx}^{\mathrm{3}} +... \\ $$$$\mathrm{find}\:\mathrm{p}\:,\:\mathrm{q}\:\mathrm{and}\:\mathrm{k}.\: \\ $$

Answered by jagoll last updated on 30/Mar/20

Commented by john santu last updated on 30/Mar/20

wow...super

$$\mathrm{wow}...\mathrm{super} \\ $$

Answered by Rio Michael last updated on 30/Mar/20

my try. let px + qx = X ⇒ (1 + px + qx^2 )^8 = (1 + X)^8 = 1 + 8x + 52x^2 + kx^3 (1+ X)^8 = 1 + 8X + ((8(7))/2) X^2 + ... = 1 + 8(px + qx^2 ) + 28(px + qx^2 )^2 +... = 1 + 8px + 8qx^2 + 28p^2 x^2 + 56pqx^3 + ... as 1= 1 8p = 8 ⇒ p = 1 8q + 28p^2 = 52 ⇒ q = 3 k = 56pq +56 ⇒ k = 224

$$\mathrm{my}\:\mathrm{try}. \\ $$$$\:\:\mathrm{let}\:\mathrm{p}{x}\:+\:{qx}\:=\:{X} \\ $$$$\:\Rightarrow\:\left(\mathrm{1}\:+\:{px}\:+\:{qx}^{\mathrm{2}} \right)^{\mathrm{8}} \:=\:\left(\mathrm{1}\:+\:{X}\right)^{\mathrm{8}} \:=\:\mathrm{1}\:+\:\mathrm{8}{x}\:+\:\mathrm{52}{x}^{\mathrm{2}} \:+\:{kx}^{\mathrm{3}} \\ $$$$\:\:\:\left(\mathrm{1}+\:{X}\right)^{\mathrm{8}} \:=\:\mathrm{1}\:+\:\mathrm{8}{X}\:+\:\frac{\mathrm{8}\left(\mathrm{7}\right)}{\mathrm{2}}\:{X}^{\mathrm{2}} \:+\:... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{8}\left({px}\:+\:{qx}^{\mathrm{2}} \right)\:+\:\mathrm{28}\left({px}\:+\:{qx}^{\mathrm{2}} \right)^{\mathrm{2}} \:+... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{1}\:+\:\mathrm{8}{px}\:+\:\mathrm{8}{qx}^{\mathrm{2}} \:+\:\mathrm{28}{p}^{\mathrm{2}} {x}^{\mathrm{2}} \:+\:\mathrm{56}{pqx}^{\mathrm{3}} \:+\:...\: \\ $$$$\:\mathrm{as}\:\mathrm{1}=\:\mathrm{1}\: \\ $$$$\:\mathrm{8}{p}\:=\:\mathrm{8}\:\Rightarrow\:\:{p}\:=\:\mathrm{1} \\ $$$$\:\mathrm{8}{q}\:+\:\mathrm{28}{p}^{\mathrm{2}} \:=\:\mathrm{52}\:\:\Rightarrow\:{q}\:=\:\mathrm{3} \\ $$$$\:{k}\:=\:\mathrm{56}{pq}\:+\mathrm{56}\:\Rightarrow\:{k}\:=\:\mathrm{224} \\ $$

Commented by jagoll last updated on 30/Mar/20

good..

$$\mathrm{good}.. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com