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Question Number 103130 by chamkunda last updated on 13/Jul/20

$$$$$$findoutthefourthmemberofthe$$$$followingformulaafterexpansion$$$${[\pi x+\frac{2}{x}]}^8$$

Commented by chamkunda last updated on 13/Jul/20

$$pleasecansomeonehelpandanswerthis$$$$findoutthefourthmemberofthe$$$$followingformulaafterexpansion$$$${[\pi x+\frac{2}{x}]}^8$$$$$$$$findoutthefourthmemberofthe$$$$followingformulaafterexpansion$$$${[\pi x+\frac{2}{x}]}^8$$

Answered by floor(10²Eta[1]) last updated on 13/Jul/20

$${(\mathrm{a}+\mathrm{b})}^{\mathrm{n}}=\underset{\mathrm{k}=0}{\sum ^{\mathrm{n}}}\left(\frac{\mathrm{n}!}{\mathrm{k}!(\mathrm{n}-\mathrm{k})!}\right){\mathrm{a}}^{\mathrm{n}-\mathrm{k}}{\mathrm{b}}^{\mathrm{k}}$$$${(\pi \mathrm{x}+\frac{2}{\mathrm{x}})}^8=\underset{\mathrm{k}=0}{\sum ^8}\left(\frac{8!}{\mathrm{k}!(8-\mathrm{k})!}\right){\left(\pi \mathrm{x}\right)}^{8-\mathrm{k}}{\left(\frac{2}{\mathrm{x}}\right)}^{\mathrm{k}}$$$$\mathrm{the}\mathrm{fourth}\mathrm{member}\mathrm{is}\mathrm{when}\mathrm{k}=3:$$$$\stackrel{\mathrm{k}=3}{\Rightarrow }\frac{8!}{3!5!}{\pi }^5{2}^3{\mathrm{x}}^2=448{\pi }^5{\mathrm{x}}^2$$$$$$

Answered by bemath last updated on 13/Jul/20

$${(\pi x+\frac{2}{x})}^8=\underset{n=0}{\sum ^8}{C}_n^8{\left(\pi x\right)}^{8-n}{\left(\frac{2}{x}\right)}^n$$$$=$$$${\left(\pi x\right)}^8+8{\left(\pi x\right)}^7\left(\frac{2}{x}\right)+28{\left(\pi x\right)}^6{\left(\frac{2}{x}\right)}^2+56{\left(\pi x\right)}^5{\left(\frac{2}{x}\right)}^3+...$$

Commented by 1549442205 last updated on 13/Jul/20

$$\mathrm{I}\mathrm{agree}\mathrm{to}\mathrm{this}\mathrm{expansion}{\mathrm{a}}_3=\frac{8!}{2!6!}{\pi }^6{\mathrm{x}}^6\times \frac{4}{{\mathrm{x}}^2}$$$$=112{\mathit{\pi }}^6{\mathbf{x}}^4,\mathbf{excuse}\mathbf{me}{\mathbf{a}}_4=448{\mathit{\pi }}^5{\mathbf{x}}^2\mathbf{as}$$$$\mathbf{above}$$

Commented by floor(10²Eta[1]) last updated on 13/Jul/20

$$\mathrm{he}\mathrm{wants}\mathrm{the}\mathrm{fourth}\mathrm{member}\mathrm{not}\mathrm{the}\mathrm{third}$$$$\mathrm{one}$$

Answered by OlafThorendsen last updated on 13/Jul/20

$${\mathrm{C}}_8^3{\pi }^5{x}^5\frac{2^3}{x^3}=\frac{8!}{3!5!}{\pi }^{5}8x^2=448{\pi }^5{x}^2$$

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