If 1+2x+3x210=a0+a1x+a2x2+…+a20x20 then a2a1=

(1+(2x+3x2))10=1+C1 10 (2x+3x2)+C2 10 (2x+3x)2+...... a2=Coeff x2=C1 10 3+C2 10 (4) =30+180=210a1=Coeff of x=(10) (2)=20a2a1=21020=212=10.5

If 1+2x+3x210=a0+a1x+a2x2+…+a20x20 then a2a1=

(1+(2x+3x2))10=1+C1 10 (2x+3x2)+C2 10 (2x+3x)2+...... a2=Coeff x2=C1 10 3+C2 10 (4) =30+180=210a1=Coeff of x=(10) (2)=20a2a1=21020=212=10.5

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If ${(1 + 2 x + 3 x^{2})}^{10} = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{20} x^{20}$ then $\frac{a_{2}}{a_{1}} =$

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Detailed Solution

$\begin{array}{rcl} (1 + {(2 x + 3 x^{2}))}^{10} & = & 1 + {}^{10}{C_{1}} (2 x + 3 x^{2}) + {}^{10}{C_{2}} {(2 x + 3 x)}^{2} + . . . . . . \\ a_{2} & = & Coeff x^{2} = {}^{10}{C_{1}} 3 + {}^{10}{C_{2}} (4) \\ = & 30 + 180 = 210 \\ a_{1} & = & Coeff of x = (10) (2) = 20 \\ \frac{a_{2}}{a_{1}} & = & \frac{210}{20} = \frac{21}{2} = 10.5 \end{array}$